Fiction by definition requires a lie, the scale and complexity of that lie will set a ball park for the amount of mental gymnastics required, nonetheless all fiction will at some point produce something that can't be made to add up precisely because of that premise.
By "lie" I assume you mean a departure from the true or real world. But this could be quite minor, like a simple name change. In this trivial example of fiction, squaring the circle (a euphemism for "impossible task," I assume, and not some more transcendental requirement) isn't a necessity, and so there is no "definition" of fiction that requires accepting impossibilities. But at this point, this seems more a semantic argument and probably isn’t an actual disagreement. I agree the more complex the fiction is, the more departures from the real world it will have, and therefore the greater the likelihood unseen or unforeseen elements are or might become inconsistent with one another, or in some other ways, inconsistent with the fictional universe's premises (which are givens, but allowed to be inconsistent with the real world without violating "internal" consistency - like the existence of warp drive and FTL travel is a given, and inconsistent with the Real World, AFAIK, but not inconsistent wit the Trek Universe).
Sci fi just tends to require bigger lies, you cannot deviate from reality to the extent star trek does and still maintain complete consistency, the question is how much does it matter to us when the inconsistencies stand out. For me, not much at all.
I'm not sure it can't be done at all, bearing in mind the huge difference between consistency with the real world, and internal consistency with itself and its stated or later accepted premises. I do agree, however, Trek failed in many regards, but that's not by definition, IMO, or a requirement of fiction, science fiction or fiction in general, but more a natural product of lax oversight and numerous contributing authors and even budget and time constraints producing the product in a manner that would not allow one to "properly" depict something as intended, but only as the budget or time allowed, which might depict some inconsistencies. But even there, the creators did strive for consistency on core matters wherever possible, and this was by design, lest one wouldn't be playing in the same sandbox. Also, writing it as they went in the beginning didn't help.
Regardless of how they arise, many inconsistencies matter a great deal to me when I perceive them to be a product of laziness or ignorance, or when the logical ramifications that would naturally follow from them would become quite problematic for future episodes. There are many other inconsistencies, however, that are virtually meaningless and certainly hardly worth consideration beyond a moment's thought, if that.
Trek was never really intended to be about world building to the extent much of the fan base expect, it was more in line with a series of morality plays which utilize the same basic background in order to facilitate the practical questions of making a TV show and to engage the viewer with a familiar set of characters and basic setting. What primarily mattered, however, was what happened next, what the actions of those characters and the events they found themselves part of meant to the viewer, how they resonated with the real world of racism, sexism hot and cold wars.
And not to discount those important contributions to modernity, but the reason Trek (and many other fictional universes) have or develop such followings is because these worlds capture the imagination, not just what they did, but what else might happen there, or here, given those possibilites. And by discussing it, fans find they can increase their enjoyment and entertainment value and maybe even the social worth in the real world by playing in the same playground with others or taking to heart some lessons they saw. But like most games, things run better when everyone knows and obeys the rules.
Otherwise, without such a fan following and appeal to something greater than perhaps intended, it would likely just be a good show from the late 60's but hardly have such a huge following today, or the topic of conversation here, let alone the inspiration for the entire social media board.
All this insistence on canon and consistency is largely external in that sense, a burden the original creators never really foresaw the show having to bear fifty years later. TOS wasn't even remotely consistent from episode to episode, yet for some reason people insist on tearing holes in later shows for sitting shakily on those foundations.
I'd say it did a fairly good job at being consistent in most areas where it mattered, allowing for inconsistencies due to budget and time constraints and simple mistakes to be retroactively reconciled where necessary, or explained where possible. The original creators may not have foreseen this show's popularity 50 years later, but what you claim is a burden is also largely the reason anyone is still talking about it to this degree 5 decades on.
There are many ways to love Trek, and it takes them all to support the Trek Phenomenon. Luckily, it is never a requirement individuals must love all aspects of Trek to partake in the parts they care about it, but hopefully nobody would make the mistake of thinking if another doesn't love Trek in the same way they love Trek, then those others are not loving it correctly or in the only meaningful way.
If that's done for fun, great, more power to you, but where I think it becomes a problem is the extent to which people become invested in it, become angry about later inconsistencies to the point they will consider a show worthless or personalize that criticism against an individual in the creative team. You don't have to look far even within these boards to find people who seemingly spit bile at any mention of the people-making Disc. In some cases it's not hard to imagine their wishing them harm. Writers and producers have in the past faced torrents of abuse and even threats from "fans" because of this stuff, fans who ironically didn't seem to realize the fault lay with them, they had missed the point.
This is doubtlessly true, and the bane of many things other than Trek, as well. I swear, I see it everywhere in social media today.
I like this. Yeah? Well then you suck, you idiot. I love that for this reason. You're kidding me. I mean, yeah, love that, but for THAT reason? What are you, some loser?
It has all the charm and relevance of an argument that if somebody's favorite flavor of ice cream isn't the same as your favorite flavor, then they are in some way a defective human being or they don't know what they're talking about. It makes no sense. So I would never support another's claim their personal preferences are the only true or correct or proper ways, and I try to avoid giving anyone that impression if I feel passionately enough about some aspect of a fictional work to talk about it that they should, too, lest they be of questionable intelligence or of lesser moral fortitude or their opinion only holds relatively minor social worth.
But most of that problem is about the maturity of the individuals, or even the way social e-media allows anonymous people to be far less social and polite than they would in real life and in person, and should never be blamed on the particular subject matter, IMO.