I see fairly high to high positive numbers on both Dicsocery and Doctor Who (the switch to female year). Now, if you go with your premise of avoiding high positive numbers you may well miss out. I could easily attribute the difference between critic and audience scores to the fact that Discovery has gay characters, which some folks would give a negative review on for nothing but that, and for Doctor Who, butthurt people who didn't like the switch to a female doctor.
Star Trek in the past did never shy away from being controversial up to the breaking point, you've just got to remember that the first interracial kiss ever being shown on American TV screens back then in the 60s was William Shatner kissing Michelle Nichols - so Kirk kissing Uhura. This was far more risky and controversial compared to putting nowadays an openly gay couple on the ship were the reaction of most is not "WTF?" but "Finally they did it!"
It's not so much about suddenly having the first openly gay pair ever, but more about that for many people the attention of having a diverse cast and telling about it had become a tad too much, while neglecting the story at the same time. The cast of TOS back then was also diverse for its time, having black and white Americans and Russians peacefully on a star ship, exploring space together. You've got to remember this was the peak of the cold war back then, so the idea of having a Russian crew member was nothing short but revolutionary.
A series thrives on acting and story telling, so its cast, character development and stage design. The sexual preferences/gender identity of their roles is for many series a nice addon, but not the main focus which does thrive a series. You can have a successful series on a very tight budget, if the cast is acting quite well and the story telling is on par with that, and also the stages do look convincing enough for the purpose of that series. A stage does not necessarily be the fanciest thing to look at ever, if it is convincing enough and gets the job done.
A prime example of such a series for me is "Babylon 5" or, much younger, "Money Heist" from Spain. Money Heist season ond and two has a great cast, good story and good enough stages, although it showed that this was not the biggest budget production around and so the best they managed to do on that budget. It didn't matter, because cast and story carried the whole series. The same applies to Babylon 5 - graphics effects and stages were clearly on a budget, but it had an amazing cast and story.
Now let's move over to "Star Trek: Discovery." First problem is that is bears the name Star Trek, and Star Trek has a very dedicated hard core generation spanning fan base. Granddads passed it over to dads, which passed it over to childrens. Gene Roddenberry had a very optimistic and quite detailed vision of the future, which is something in such apocalyptic times we are living in right now many people do indeed enjoy for entertainment. And this is what the old fans the franchise do expect from Star Trek: a united, somewhat peaceful, humble and honest man kind, which was able to overcome the old obstacles of the pass, now the glue of the known galaxy.
And this is where Discovery differs a lot from the earlier series: it clearly shows a struggling, desperate mankind which is not above from genocide in order to ensure its own survival. It is a far darker, less optimistic more pessismistic view of the possible future, with a nittier, gritter man kind. This is the first bitter pill die hard fans had to swallow, turning Roddenberry's vision around 180 degress and many didn't like that pill.
The next thing which alienated many people is that the authors seem to have been quite a long time more focused on communicating everybody how diverse their cast is instead of telling a good story. The main focus should be a good story, if this carries well enough you can also have a very much diverse cast and not many would subject at all. But having the most diverse cast ever doesn't automatically make a good story, and telling everybody how much diverse your cast is while having only a mediocre story is a recipe for trouble, where the friendlier people will just tell you to get your priorities right again, and the naster ones will move on to far deeper obscenities. That's the problem.
Star Trek was never known for the most logical stories, it always was suffering from technobabble and deus ex machina events. But it had some great stories anyway, like "Chain of command" in TNG. Also most Star Trek series were on a tight budget.
Discovery on the other hand has an enormous budget, and it shows - the show just oozes visual effects every where, the location shots and stages are doing the rest. But the story is severely lacking, it has logical plot holes of such gigantic proportions that even most hard core fans, which normally are willing to overlook these, really cannot do that any longer. And Discovery feels more like action trek like Star Trek most of the time as well, also quite unlikeable main characters (not actors), like Michael Burnham.
And this is what pissed many people off - the re-invention of Star Trek in a very much non-Roddenberry type of way. That's the worst thing you can possibly do when you got such a generation spanning hard core fan base like Star Trek has. You might win new fans with that type of approach, while pissing off the old ones which kept the franchise running for a long time and making lots of money.
Much of this can as well be applied to the new Picard series, the whole main plot is just really, really bad and disappointing.
