Put me down as one of those people who doesn't think a Trek reboot is needed, and we can tell plenty of stories within the existing Trek canon. However, there are a lot of things we could potentially "correct" if there was a well-done reboot which would help to create a show more attuned to modern sci-fi norms. Here would be my list.
- No universal translator. I realize it's a great plot contrivance, but it makes no sense at all, because such a device would essentially need to read everyone's mind in order to work properly. Plus it never makes sense as depicted on the screen, given peoples' lips move in sync with the translated dialogue, not their own tongue. This could limit the type of stories which are told within the Trekverse, but honestly the lame first-contact style stories where we discover aliens exactly like us in every way except for one little thing needs to die.
- No transporters either. Technically speaking a transporter isn't as out there of an idea, as breaking down someone atom-by-atom and rebuilding them somewhere else probably would be feasible with a lot of processing power. But the tech originally was developed due to TOS budgetary limitations - not being able to show shuttlecraft every week. There's no reason for it in the modern era.
- Replicators are fine as is, because honestly they are just a further elaboration of modern-day technologies like 3D printing. However, a Trek reboot would hopefully have a better handle on them. For example, perhaps replicators can make anything provided they have access to a supply of the base element, meaning some rare minerals still have value. There should be a more consistent understanding of how replicators result in a post-scarcity economy as well.
- Trek's computer systems are often startlingly dumb, considering their computers should be centuries more advanced than our own. We know true AI is possible, although 24th century computers seldom break through this barrier. Even if self-aware AI is difficult to achieve, the on-board computers should be able to do a large proportion of the work of running the ship itself - everything from performing evasive maneuvers to locking weapons to engaging in repairs via semi-autonomous drones. Honestly from a story perspective this would be better, as it would free up our crew do do more away missions, and less tech the tech to plot the plot.
- Humanoid races should be dialed way back. I am fine with keeping some of the established races as humanoid, but going all the way back to TOS and some of the early movies we saw a fair number of non-humanoid races. Trek needs more of them - particularly more who are not some sort of hyper-evolved "energy being."
- Human-alien hybridization started of course with Spock, but it got to be ridiculous during the Berman era, with virtually every species shown to be cross-fertile despite having very different biochemistry. If we want to leave this in for select races, that's fine, but some in-story explanation, like the Iconians harvesting primitive humans and spreading them in modified form across the galaxy should be used.
- Starfleet needs to cease to be 90%+ dominated by humans. The Federation was always purportedly an equal partnership between many different alien races, yet virtually every time we see a Starfleet vessel 90%+ of the people onboard are human. I understand this started in part because the idea of Starfleet as being a multi-species organization developed over time, and also due to budgetary limitations. Still, either canonically find some way to explain humans being dominant, or don't show humans as being dominant.
- Similarly, Trek needs to depict the full scale of 23rd/24th century humanity. This means we shouldn't just see the Federation as "Space America" by way of whoever is easiest to cast as extras in California or Canada. Just sheer luck of the draw should mean the majority of humans in Starfleet (presuming current population trends continue) would be East or South Asian.
- In addition, Trek needs to depict far more multiracial humans. If humanity put racism behind them, one would presume that any stigma about interracial marriage had vanished. Obviously there would still be a lot of remote places on Earth where most everyone was from one background or another, but we know that a lot of Starfleet officers follow in their parents footsteps, suggesting there should be a disproportionately high level of people from backgrounds so mixed they're no longer any one thing in particular.
- I'd like a broader depiction of advanced alien races more broadly. In the Trekverse, it seems like nearly every alien race is either less advanced than the Federation, or within a few hundred years of development. Otherwise, they're just some godlike energy being. But the universe is 13.7 billion years old, and there should be many truly ancient civilizations which have not advanced into a higher plane yet are still as far beyond the Federation as the UFP is from a stone-age tribe. This would require developing a different sort of story, but it would offer up a great deal of story potential, given it would drive home that the galaxy is a huge place full of strange wonders built by the elder races.