Boring Answer coming up....
Kirk told Gillian in TVH that he was from the Late-23rd Century. "Okay, the truth. I am from what on your calendar would be the late-23rd Century... " At the end of the movie, Gillian tells him she has "300 years of catchup learning to do."
In "The Neutral Zone", Data says, "It is the year two-thousand three-hundred sixty-four." 2364 being 78 years after TVH being in the late-23rd Century checks out. And those two dates became the ones that Michael and Denise Okuda based everything else off of.
I think most of us put TOS in the 22nd century before TNG aired. Then it got retconned.
I was a little kid and not yet a fan, during the time you're talking about. BUT, if I were older and had been a fan, I think hearing from Decker that Voyager 6 was launched "300 years ago" and then TWOK saying right upfront at the beginning with big, bold letters
"In the 23rd Century... " would've solidified in my mind that Star Trek took place in the 23rd Century.
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I hate bringing up James Dixon but even
he placed TOS in the 23rd Century. He placed the beginning of TOS in 2260 because, in "Miri", Spock said the planet resembled Earth "circa 1960" and it was said that "circa 1960" was "300 years ago". Spock was clearly rounding off about 1960 and people usually round off when talking about hundreds of years ago, so 2260 wasn't solid but that's not the point. The point is "Mr. I'm Obsessed With the Chronology!" also placed Star Trek in the 23rd Century. And he's about as extreme about this stuff as you can get.