When did TOS itself indicate it happened?
PART ONE: EPISODE DATA
"Menagerie Part 1":
So Spock claimed that the events in "The Cage" happened sometime between 13.00 and 13.99 BM (Before Menagerie).
When do "Menagerie Part 1" and "The Cage" indicate that the Columbia crashed on Talos IV?:
Since Spock says "approximately eighteen years" that should be about 17.00 to 19.99 years, meaning that the S.S. Columbia crashed about 30.00 to 33.98 BM.
And:
If the crew of the the S.S. Columbia left Earth an unspecified time before crashing about 30.00 to 33.98 BM and didn't know about the time barrier being broken and faster ships being built, the breaking of the time barrier might have happened anytime after they left and last heard from Earth, and thus possibly before or after their crash about 30.00 to 33.98 BM. Since the Enterprise exists at the time "The Cage" 13.00 to 13.99 BM, it should be at least 13.00 years old in "Menagerie" and possibly over 30 years old.
And there is little clue how many centuries in the future this may be.
"Where No Man Has Gone Before" indicates the possible use of one, two, or three different year counts:
To me this indicates that "Where No Man Has Gone Before" happens between 100 and 200 years after "Nightingale Woman" was written by Tarbolde in the year 1996 TM (Tarbolde-Mitchell date), and thus sometime between 2096 and 2196 TM.
Kirk says:
This implies that Earth has had interstellar travel for at least 200 years, and thus since at least as early as some time during the period of 1896 to 1996 TM.
The service records of Gary Mitchell and Elizabeth Dehner say they were born in Eldman in 1087.7 and in Dellman in 1089.5, respectively. Since they looked to be between the ages of 20 and 40, if their birth dates were given in years the year of "Where No Man Has Gone Before" should have been sometime between about 1109.5 and 1127.7 ED (Eldman-Dellman dating).
If the second number on Kirk's tombstone was the year of Kirk's death, "Where No Man Has Gone Before" would happen in the year 1313.7 KT (Kirk Tombstone dating).
In "The Corbomite Maneuver" Kirk claims, accurately or not, that Earth has had interstellar travel for two centuries:
If Kirk is telling the truth "The Corbomite Maneuver" should be at least 200 years after AD 1966 and thus in or after AD 2166.
In "Miri" They beam down to an Earth-like abandoned town:
This would indicate that the town resembled an Earth town about AD 1950-1970, and that the planet would have resembled Earth about AD 2150 to 2470 if the disaster hadn't happened, which says nothing about the Earth year of "Miri".
In a building, they find a piano:
Which may mean the piano looks like ones made on Earth about 300 years earlier, thus putting "Miri" about AD 2150-2370. or else that it seems to have existed for about 300 years, thus being meaningless for dating "Miri".
In "The Squire of Gothos":
Since Trelane knows of Napoleon, crowned in AD 1804, and of Alexander Hamilton's death in AD 1804, if Gothos is about 800 to 1,000 light years from Earth, the date of "The Squire of Gothos" would be in or after AD 2604 to AD 2804, if Jaeger and Kirk's reasoning is correct.
But since Trelane can move his planet Gothos in space, and since Trelane knows Earth words and can speak Earth languages, it seems very likely that Jaeger and Kirk were wrong about Trelane's source of information about Earth, and thus it is not a valid clue to the date of "The Squire of Gothos".
In "Shore Leave" Sulu finds a pistol on the planet:
Assuming that in the alternate universe of Star Trek they stopped making police special pistols about AD 1900 to 2050, and assuming that Sulu's couple of centuries means one to two centuries, "Shore Leave" would be about AD 2000 to 2250.
In "Tomorrow is Yesterday" they travel back in time to Earth:
MAN [OC]: This is the five thirty news summary. Cape Kennedy. The first manned Moon shot is scheduled for Wednesday, six am Eastern Standard Time. All three astronauts who are to make this historic
(Kirk signals it cut off)
KIRK: Manned Moon shot? That was in the late 1960s.
SPOCK: Apparently, Captain, so are we.
So apparently the first manned moon shot happens in the late 1960s - between 1966.66 and 1969.99 TY (Tomorrow is Yesterday dating) in the alternate universe of Star Trek. Later:
FELLINI: I am going to lock you up for two hundred years.
KIRK: That ought to be just about right.
Which, if taken seriously, implies that part of Kirk's lifetime will be sometime between about 2066.66 and 2269.99 TY.
In "Space Seed" Spock says:
Therefore, "Space Seed" should happen sometime between 2090 SS and 2999 SS (Space Seed dating).
So faster than light interstellar travel did not start until about 2016-2020 SS, or possibly much later. Since the S.S. Valiant traveled interstellar space 200 years before "Where No Man Has Gone Before", "Where No Man Has Gone Before" should happen in or after 2216-2220 SS.
So Kirk and Co. guess than Khan left Earth sometime in the early or the mid 1990s SS, and thus between 1990.00 and 1996.66 SS.
So Kirk states that Khan left Earth 200 years earlier putting "Space Seed' in about 2090.00 to 2296.66 SS. Since the Valiant left Earth in or after 2016-2020, "Where No Man Has Gone Before" should happen in or after 2216-2220 SS and "Space Seed" should happen in about 2216-2296.66 SS.
In "Metamorphosis" they meet Zefram Cochrane:
Spock doesn't correct McCoy, suggesting that Cochrane disappeared sometime between 125 and 175 years earlier, or at the most between 100 and 200 years earlier. So Cochrane should have disappeared sometime between 125 and 175 PM (Pre Metamorphosis dating) or possibly sometime between 100 and 200 PM.
So if Cochrane was between 87.00 and 87.99 years old when he was lost in space, he should have been born sometime between 262.99 and 212.00 PM, or at the most sometime between 287.99 and 187.00 PM.
If Cochrane was between twenty and forty when he "discovered the space warp" that should have been between 242.99 and 172.00 PM, or at the most between 267.99 and 147.00 PM.
In "Wolf in the Fold" the computer is asked about the meaning of "redjac":
This implies that if Hengist and Tark are correct, "Wolf in the Fold" should be about 200 to 1,000 years after AD 1888 when Jack the Ripper killed, and thus sometime about AD 2088 to AD 2888.
If Jack the Ripper killed in AD 1888, and also during the 19th century according to the WF dating system (Wolf in the Fold dating) the year one in the WF dating system would be sometime between 12 BC and AD 88.
Dates given in the WF system include:
The latest date is 2156 WF, which would be sometime between AD 2144 and AD 2244. Thus "Wolf in the Fold" should happen sometime between AD 2144 and AD 2888.
This implies that "Wolf in the Fold" is probably happening years or decades after 2156 WF.
In "Requiem for Methuselah" Mr. Flint turns out to be unusually old:
If Flint correctly converted his birth year to modern reckoning, 3834 BC can be converted to a date in the Holocene Calendar or Human Era reckoning by subtracting 3834 from 10,000 to get 6166 HE.
If Flint was born between 6166.00 and 6166.99 HE, and if he was 6,000.00 to 6,000.99 years old in "Requiem for Methuselah", the date would be between 12166.00 HE and 12167.98 HE. Converting those dates to Anno Domini by subtracting 10,000, "Requiem for Methuselah" should be between AD 2166.00 and AD 2167.98.
If Spock's estimate of Flint's age was accurate to within 250 years, "Requiem for Methuselah" should be between 11916.00 HE and 12416.99 HE, or AD 1916.00 to AD 2416.99.
If Spock's estimate of Flint's age was accurate to within 500 years, "Requiem for Methuselah" should be between 11666.00 HE and 12666.99 HE, or AD 1660.00 to AD 2666.99.
If Spock's estimate of Flint's age was accurate to within 1,000 years, "Requiem for Methuselah" should be between 11166.00 HE and 13166.99 HE, or AD 1166.00 to AD 3166.99.
And if Spock's estimate of Flint's age was not precise to within 1,000 years Spock would probably have said something like Flint's age was on the order of five to seven thousand years.
In "The Savage Curtain" briefing room scene:
So when Scott gestured for emphasis Spock corrected him about the present direction to the planet Earth, which would be constantly changing as the Enterprise orbited Excalpia. Spock corrects people for not keeping track of the ever changing direction to Earth. But Spock didn't correct Scott for any error in the much simpler calculations needed to state how long ago Abraham Lincoln died.
Abraham Lincoln was shot on April 14, 1865 and died on April 15, 1865. Which is roughly about AD 1865.2814 to AD 1865.2876. Adding 300.00 to 300.99 years gives a date of AD 2165.2814 and AD 2166.2776.
If Spock would have accepted Scott's "three centuries" if it was accurate within 25 years, "The Savage Curtain" would happen sometime between AD 2140.2814 and AD 2190.2876.
If Spock would have accepted Scott's "three centuries" if it was accurate within 33.33 years, "The Savage Curtain" would happen sometime between AD 2131.9514 and AD 2198.6176.
If Spock would have accepted Scott's "three centuries" if it was accurate within 50 years, "The Savage Curtain" would happen sometime between AD 2115.2814 and AD 2215.2876.
If Spock would have accepted Scott's "three centuries" if it was accurate within 75 years, "The Savage Curtain" would happen sometime between AD 2090.2814 and AD 2241.2876.
If Spock would have accepted Scott's "three centuries" if it was accurate within 100 years, "The Savage Curtain" would happen sometime between AD 2065.2814 and AD 2265.2876.
But I find it very hard to believe that Spock would not have correct Scott about the time since Lincoln's death if it was actually less than 200 years or more than 400 years. So "The Savage Curtain" should have been sometime between AD 2065.2814 and AD 2265.2876.
PART TWO: SUMMARY
So several TOS episodes give a date range in unspecified dating systems. Any of those unspecified dating systems could be Anno Domini dating, but since they are not specified there is no way to be certain about that.
A few TOs episodes do give evidence to estimate a date range in Anno Domini dating.
If the evidence is interpreted one one way, "Miri" might happen sometime about AD 2150-2370, but the evidence can be interpreted as not implying a date range for the episode.
"The Squire of Gothos" might happen in or after AD 2604 to AD 2804 if interpreted one way, but can be interpreted as having no evidence for its date range.
Kirk's quip in "Tomorrow is Yesterday", if taken seriously, implies that part of Kirk's lifetime will be sometime between about 2066.66 and 2269.99 TY. and if Kirk & Spock use Anno Domini dating some part of Kirk's lifetime will be sometime between about AD 2066.66 and AD 2269.99. If those assumptions are correct.
Assuming that in the alternate universe of Star Trek they stopped making police special pistols about AD 1900 to 2050, and assuming that Sulu's couple of centuries means one to two centuries, "Shore Leave" would be about AD 2000 to 2250. This seems like the best range so far, though based on two assumptions.
"Wolf in the Fold" should be about 200 to 1,000 years after AD 1888 when Jack the Ripper killed, and thus sometime about AD 2088 to AD 2888, if Hengist and Tark are correct about how long ago that was.
"Wolf in the Fold" should also be after 2156 WF, and thus after sometime between AD 2144 and AD 2244.
"Requiem for Methuselah" should be between 11166.00 HE and 13166.99 HE, or AD 1166.00 to AD 3166.99.
"The Savage Curtain" should have been sometime between AD 2065.2814 and AD 2265.2876.
Could you expand on this please?