To quote TREK novelist Christopher Bennett:
Fans have always had to squint and gloss over the differences of interpretation in order to pretend that these works of fiction created by different people with different ideas could represent a consistent reality. If you want to be that obsessively nitpicky, then you'll have to admit that 'Where No Man Has Gone Before' is set in an alternate universe where Kirk has a different middle name, 'Mudd's Women' is set an alternate universe where they use lithium instead of dilithium, most of TNG's first season is set in an alternate universe where Data used contractions and showed emotion, etc.
STAR TREK has never, ever, EVER been an actually consistent reality. We only choose to pretend it is by ignoring or rationalizing the hundreds and hundreds of contradictions it already contains. So either you're willing to suspend disbelief and play along with the pretense that there's a single universe, or you're not and you have to admit that there are countless mutually contradictory versions of Trek already.
To claim that previous TREK is completely reconcilable but the newest thing is completely irreconcilable is a self-contradiction.
Roddenberry's take: TOS was an imperfect dramatization of the crew's adventures and that later TREK productions were able to come closer to getting it right. It wasn't the TREK universe that was changing, just the way in which it was dramatically recreated for 20th-century television viewers.
Some of my favourite inconsistencies:
The original series took almost half a season to pin down the 23rd century era, with the time period referred to as the 21st or 28th century. Kirk at one point says he works for the United Earth Space Probe Agency before it became Starfleet and Earth became the Federation. Spock is emotional in the early episodes and made a rape joke. Kirk's initial in the first episode produced is "R." Spock refers to his parents in the past tense, but they guest-star later on. McCoy says that the "Vulcanians" were conquered by Earth.
From a production standpoint, the Starfleet arrowhead was meant to be for all starship crews, but for a number of TOS episodes, costumers misunderstood "Charlie X" in which the crew of the Antares had their own insignia (as merchant marines) and took that to mean each ship had its own individual badge when designing costumes. DISCOVERY uses the triangular symbol as intended rather than as it was onscreen.
With TNG's early seasons, Picard was a cruel leader prone to putting his people in difficult situations just to screw with them, Data was emotional, Troi experienced other people's emotions rather than being aware of them, Worf was animalistically feral, Starfleet regularly vacationed on pre-warp planets, holodeck matter existed outside the simulator, the Borg ignored organic life -- none of which was retained as Picard became gentle to the point of babying Barclay, Data became emotionless, Troi's powers dialed down, Worf became smart, the Prime Directive became much stricter and the Borg started assimilating people.
The mannered and bizarre Ferengi of TNG's Season 1 are not the capitalist caricatures of DS9, the makeup for Trills in TNG was ignored in DS9, Voyager travelled back to the 1990s where the Eugenics Wars, established in TOS, are not present or mentioned.
FIRST CONTACT and ENTERPRISE have warp drive in the 22nd century, but TOS' "Balance of Terror" established that the Earth-Romulan War unfolded at sublight speeds. TNG had Wesley depart Starfleet to ascend to higher planes of existence with the Traveler; DS9 had Worf become a Klingon ambassador by the finale "Nemesis" has him -- yet NEMESIS shows both back in Starfleet.
In "Operation: Annihilate!," Kirk's brother, Sam, is killed. Yet, in STAR TREK V, Kirk remarks, "I once lost a brother. I was lucky to get him back," referring to Spock and suggesting that Kirk has forgotten he had a sibling who died with his wife and left behind an orphaned nephew.
STAR TREK has never been a documentary. But if you must have an in-universe blanket explanation, the simplest route is that Data's trip to 19th century Earth in "Time's Arrow" and the time travel of FIRST CONTACT along with the Temporal Cold War of ENTERPRISE have caused some details of TOS to shift and some of the contradictions are due to the time travel ripples taking effect.
The novels and comic books, however, tend to offer rationalizations via new stories that weren't aired on TV or shown in theatres. The comic book adaptation for STAR TREK V amended Kirk's line about his brother to say that Kirk lost "two brothers" and was lucky to get "one back." My personal explanation for the error: Sam Kirk was probably, in an untelevised story, resurrected due to some VOYAGER-esque time travel rewind that retroactively erased his death.
And maybe there are many variants of Ferengi and Trills, we were seeing Picard during periods of indigestion during Season 1, Troi mastered her psi-powers, Data was experimenting with simulated emotion, the Earth-Romulan War unfolded in areas of space where warp drive couldn't be used, Worf got counselling, etc..
There's some stuff that's best ignored, however. It's grossly out of character for Spock as he took shape to joke that a woman who was sexually assaulted by an evil double of Captain Kirk enjoyed the experience. It is outrageous to claim, as TOS did, that no woman has ever captained a starship.
I prefer to simply think that these events didn't happen, much like Quinn shrugging off Wade being in a rape camp or spending a season finding coordinates to Kromagg Prime and a way to bypass the Slidecage only to blow both off in "Revelations."
And I don't think we need to restrict STAR TREK to technology that was feasible to render on TV in the 1960s; the show should reflect a future based on the world we have today. And on the level of TV production, there is really insufficient time to worry about it at all.
In a podcast, "Desperate Hours" author David Mack said that he read the TV scripts, passed along any contradictions he didn't think could be reconciled and some were amended and some weren't. He added that he offered the TV producers three paths for "Desperate Hours": he could describe the 60s Enterprise as being visually in line with 2017 DISCOVERY ships with holograms and jacketed uniforms and metallic surfaces. He could describe the 2017 DISCOVERY ships as being visually in line with the 60s STAR TREK with switches and dials and pastel colours.
Or he could describe both the 60s and 2017 ships exactly as they appeared onscreen -- and declare with a straight face that the 60s ships are in fact more advanced than the 2017 ships and have the characters consider the 60s style to be more futuristic than the DISCOVERY ships.
They asked him to take the third option. That said, a lot of this could be side-stepped if DISCOVERY were set in the 25th century and DISCOVERY has, for now, given no real reason why it's set in the 23rd aside from Michael Burnham being Sarek's adopted daughter.