It's not theater—it's another medium.
All acting is theater. All set design is theater. Theater has been a human art form for thousands of years. Film and television drama are a recent refinement upon it, existing for a fraction of a percent of the whole. They're a subset of theater. This was even more the case in the 1960s than it is today, because the resources for strict realism rarely existed, and audiences at the time still had functional enough imaginations that they were capable of suspending disbelief, because they had to. TOS often had stagey sets and lighting and very theatrical makeup (e.g. Spock's heavy eye shadow), because TV was only a couple of decades old at that point and its makers were still approaching it as theater.
Also, how many times are episodes of television presented to an audience again, with a different cast, using the same script?
That has actually been done from time to time, e.g. with an episode of Mannix (Desilu/Paramount's sister show to TOS and Mission: Impossible) that was made twice five or six years apart, with William Shatner as the featured guest in one of them. Then there was the 1988 Mission: Impossible revival which was conceived during a writers' strike as a remake of unaltered scripts from the original series, although the strike was resolved early enough that only 5-6 episodes were remakes and all were rewritten to a greater or lesser extent.
Besides, I was just using it as an illustration of the point that what we see onscreen is an interpretation of the underlying idea, that what really matters is the story being told and not the superficial trappings of the presentation. That principle applies whether it's a remake or not. As I said, this is how Roddenberry himself perceived TOS, as a dramatization that only approximated a vision of the future that 1960s technology and budget weren't capable of capturing exactly. In his foreword to the TMP novelization, he pretended to be a 23rd-century TV producer who'd made an "inaccurately larger-than-life" dramatization of the Enterprise's adventures and promised that TMP was a more realistic dramatization this time around.
After all, Roddenberry got his start in TV by adapting real police cases to be dramatized on Dragnet. I've realized that that informed his approach to Star Trek, another show whose episodes were framed by narration of its main character's official report describing the events therein depicted.
However, suggesting that we're watching a fictionalization or dramatization of events to handwave any inconsistency—while certainly ingenious and, well, handy—doesn't really float my boat either. I agree with @Poltargyst.
I think it was probably easier for 1960s creators and audiences to accept the idea, because American society at the time was more accustomed to live theater than we are today, and much of 1950s-60s TV was simply stage plays with cameras pointed at them. Then there was the popularity of shows dramatizing historical events, like all the WWII shows and period dramas, Walter Cronkite's You Are There where he pretended to be a TV reporter covering "news" events from hundreds or thousands of years in the past, even Desilu's The Untouchables. (Although the original Desilu Playhouse production of that was based on a "nonfiction" book that we now know to have been mostly fraudulent, and the TV series that followed it was entirely fictional.) Audiences knew that what they were seeing in such productions was not the real history but simply a dramatized interpretation of it. It's not that hard to apply the same thinking to something set in the future, which obviously cannot be depicting a real event.
And to get back on-topic, Star Trek is hip-deep in artistic license, and always has been, and if you don't like it, how could you possibly like Star Trek? Wouldn't you be better off sticking to "hard" science fiction?
Indeed. Virtually any outer-space FX shot in Star Trek cannot possibly be taken as a literal depiction of events, starting with the shots in TOS where the Enterprise followed a visibly curved path around a planet even though its path at a high orbit would be many times flatter than the apparent surface of the planet when you're standing on it -- and culminating in the frequently nonsensical CGI shots in the modern Secret Hideout shows that often directly contradict the dialogue (like putting a space station in Earth orbit when the dialogue said it was 100 AU from Earth, or depicting a turbolift shaft as a high-tech thrill ride in a hammerspace too large to fit into the ship). See also any shot where a weapon beam is visible in vacuum, or where two ships are shown in close visual range when dialogue says they're tens of thousands of kilometers apart. Or ships brightly lit in the depths of interstellar space, ships always being right-side-up relative to each other, sound in space, etc.
I have a stump speech about television and movies being audio-visual storytelling mediums. The words, the actions, the sets, the visual effects . . . everyone's efforts come together to create the complete tale we see and hear and experience. Star Trek has taken pains to try to present its episodes as stories that are real within the universe it inhabits. Otherwise the money to build the orbital drydock and film scenes involving it would not have been necessary . . . otherwise the holodeck recreation from "Relics" or the Defiant bridge from "In a Mirror, Darkly" could've been created in any way the showrunners 'wished to visualize'.
Yes, but it's a matter of degree. It's not a binary, yes-or-no question. Roddenberry wanted Star Trek to be as believable as he could make it, but he recognized that there were practical and budgetary reasons why he could only partially achieve that, so he accepted the need to make compromises, one of the biggest ones being the preponderance of Earth-duplicate alien cultures allowing the use of standing backlot sets, historical costumes, and such. He did the best he could with what was possible, but recognized that it couldn't be perfect. And later, when TMP and TNG came along, he tried to use the improved budgets and technology to bring the fiction that much closer to what he imagined, with fewer compromises. Every work of art falls short of perfection, and all an artist can do is try to get as close to it as time and capability allow.
(As an aside, however, it may help explain the "visual reboot" / "reimagine" that is the Discoverse. When you read some sci-fi novel in the 1980s you might've imagined the setting in a rather "cassette futurism" sort of way, but if you were to re-read the novel today you might imagine it with today's tech stylings.)
I don't think everything in the Secret Hideout shows can be taken literally -- especially the VFX -- but in this day and age, I think it makes sense to treat TOS as the approximation and the later productions as closer to the truth. After all, there's no way a plausible 23rd century would look like it was built with 1960s props and set components, or have the obliviously casual misogyny that was all too common in TOS.