The problems with making Star Trek believable are multiple, connected, and hard to disentangle:
1. To create a society of non-capitalistic space explorers a number of reality breaking techs were invented.
Nah, the first supposition was that there would be a society of non-capitalistic space explorers. The technology wasn't invented to create that, the technology was added afterward, and often for no other reason than to dodge the restrictions and conventions of television production.
Take the transporters for example. They were invented simply as a means to move characters from one place to another and avoid expensive model shots of shuttles, planets, etc. Warp drive is little more than a convenient excuse to be able to put the cast in a different location every week rather than having to explain why they never covered the YEARS of travel between planets that would otherwise be required.
2. Since each of these techs was by itself reality breaking each of them had to have boundaries created, a feat which usually involved more impossible tech.
This was definitely only a problem in the Next Generation era and beyond. In the original series there was little if any tech that required building upon existing tech and NONE of it required explanation or justification. It just WAS. It was introduced for no other reason than to hang a plot onto it for an episode and then forget about it. Gene's interest was not in the invention of interesting technologies for thier own sake. He wanted to explore the human condition using a fictional future and its technologies as a backdrop. The Romulans with their cloaking device and energy plasma weapon were invented basically to repackage a submarine warfare plot. Replicators were a means of providing characters with food without having to have sets of large messhalls or cafeterias, kitchens and pantries, as well as, say, nazi uniforms, 1930's gangster suits, etc. without having to show massive storage rooms for such gear or explain their absence.
5. FTL, replicators and deflector shields (among other things) are unfixable, and the core of everything that Star Trek is.
Deflectors are just throwaway stuff. That is, you say you have FTL travel to get the crew from A to B every week. Someone says, "What about hitting single molecules at such speeds being able to destroy the ship?" You answer, "deflectors". How do they work? WHY do they work? Who CARES? Geeks like us sitting at home wanted those explanations. Gene didn't seem to feel it was necessary to justify that stuff unless and until it needed justification IN an episode for some reason.
Personally, I think the place to take an "alternate" Star Trek is to simply start to examine the Trek universe AWAY from Star Fleet and interplanetary diplomacy. See, even though Gene assumed this future society that was supposed to be all communist and egalitarian and BEYOND the struggles of humanity that we faced in the present day he sure seemed to me to be doing his level best to prove that it was ILLUSORY. Even in the original series, although the Federation is beyond racism and bigotry - the Enterprise crew is dealing with racism and bigotry both with the character of Spock and in dealing with alien races (or alien races dealings with each other). Even though the ships crew get their meals from a slot in the walls, the rest of the galaxy including Federation outposts and starbases seem to need shipments of food like quattrotriticale, or grow fruits, grains, and vegetables like humanity has done since the dawn of civilization. For all the medical strides that have been taken there still seem to be plenty of diseases, infections, birth defects, and so on that CANNOT be cured by magical hand-held-whirly-thing technologies.
Those sorts of examples show that despite the strides that had been made this future humanity IS still dealing with... its HUMANITY. People still commit crimes. They lie, cheat, steal, betray, subvert and turn traitor. Even the ones who personally don't do those things still have to deal with others who DO. Humanity may make progress but will never cease being flawed, fallable humans. THAT is what Star Trek is based on and the technology is otherwise either a mere convenience for storytelling purposes or it utterly irrelevant to it.
