Logically, if any continuity uniquely from TAS is found in any of the movies or follow on series, then TAS occurred. If not, then TAS didn't happen and not part of canon (it was just a cartoon). Example: Captain April was never mentioned in TOS, but he was introduced in TAS and later referenced in DISCO, therefore, TAS occurred. More examples will further cement it as canon. If you believe it is part of canon, and since the uniforms are the same as in TOS and not the same a few years later in TMP, then TAS is bracketed to occur either during or shortly after the TOS series. Since the stardates overlap then it is in the same 5YM timeframe.
Here are better examples than Christopher's.
James Blish often mentioned characters, technology, etc. from one story or series in another story or series. So is all of Blish's fiction happening in alternate universes in one multiverse, or are the various Blish stories unconnected? It might be just a coincidence that the method of instant communication is called a Dirac transmitter in several different Blish stories, for example.
And Blish carried on this approach in his
Star Trek writings. In his adaptations of TOS episodes he mentioned Gifford Bonner and the Vegan Tyranny from
Cities in Flight, and in
Spock Must Die! Xixiobax Jewelworms from "This Earth of Hours".
In the TAS episode "Yesteryear" Spock as a child goes to the mountains before his official survival test. In the TNG
episode "Unification Part 1":
SAREK: No. I never knew what Spock was doing. When he was a boy, he would disappear for days into the mountains. I asked him where he had gone, what he had done, he refused to tell me. I insisted that he tell me. He would not. I forbade him to go. He ignored me. I punished him. He endured it, silently. But always he returned to the mountains. One might as well ask the river not to run. (lies down again) But secretly I admired him, the proud core of him that would not yield.
This seems to refer to events in "Yesteryear". So the response to this is to say either:
1) This proves that TAS is canon with the rest of
Star Trek.
or:
2) There never was a need to prove TAS canon because TAS always was canon.
or:
3) Nothing could possibly prove TAS canon; TAS can never be canon.
or:
4) The evidence is still insufficient to say if TAS is cannon.
Memory Alpha's article on TAS has a long list of various elements of other
Star Trek productions that more or less reference TAS:
https://memory-alpha.fandom.com/wiki/Star_Trek:_The_Animated_Series
On the subject of TAS, in "The Slaver Weapon" Larry Niven adapted his story "The Soft Weapon", including elements from his Known Space series, including the Kzin and the Slavers. Thus there seems to be a degree of overlapping between story elements in TAS and in Known Space.
One would normally expect that any
Star Trek story where someone time travels to the west of the USA in the 19th century would have to be consistent with actual history of the west, just as much as it a time travel story to any other historical setting would have to be consistent with actual history. But in the case of
Star Trek it is more complicated.
Warner Brothers made several television westerns in the 1950s, and there were a number of crossovers from one series to another. In the most famous example, the
Maverick episode "Hadley's Hunters" had crossovers with
Cheyenne,
Bronco,
Sugarfoot, &
Lawman & a reference to
Colt.45 and maybe even to
Wanted: Dead or Alive. So there seems to be a
Maverick universe of related TV westerns.
In
Kenny Rogers as The Gambler, Part III: the Legend Continues (1989) Brady Hawkes gets involved in Sioux affairs and apparently prevents the killing of Sitting Bull, the Ghost Dance troubles, and the Wounded Knee Massacre in 1890, thus putting that series in an alternate universe to real history. In the sequel
The Gambler Returns: The Luck of the Draw (1991) Brady Hawkes travels to San Francisco to a poker tournament in honor of the late Mr. Paladin, a reference to
Have Gun Will Travel. Brady Hawkes meets characters from a number of TV westerns including
Bat Masterson,
Kung Fu,
The Rifleman,
The Westerner,
The Virginian,
Rawhide,
The Life and Legend of Wyatt Earp,
Cheyenne, and
Maverick. This establishes a Gambler universe of TV westerns, which overlaps with the
Maverick universe through
Cheyenne and
Maverick.
In Barbara Hambly's novel
Ishmael (1985) Spock travels back in time to the 19th century west. But not to the historic west of real history, but to the west of the television series
Here Come the Brides. And there are also references to characters in television westerns such as
Bonanza,
Gunsmoke,
Have Gun Will Travel, The Rifleman,
Rawhide, and
Maverick, as well as science fiction like
Star Wars,
Battlestar Galactica, and
Dr. Who. So the television shows in the Gambler Universe and the
Maverick universe might be real historical events in
Ishmael (1985).
And this is only the beginning of a discussion of the many fictional universes which might be considered connected to
Star Trek. See:
https://www.trekbbs.com/threads/star-trek-is-part-of-mcu.293148/
Therefore, at an absolute minimum, the official standard is that all live action
Star Trek productions are certainly canon, though some may happen in alternate universes. And many may consider TAS to be amoung the dramatic productions that are certainly canon.
And some
Star Trek fans might consider that their personal canon includes a lot more stuff like the Tommy Westphail Universe, the Wold Newton Universe, the Cthulhu Mythos, the MCU,
Cities in Flight, etc., etc., though other fans might not agree with some or all of those additions to
Star Trek canon.