Multiple instances of Starfleet knowing about things that the TOS/TNG/DS9 crew didn't or discovered on their own years after DSC, such as the Mirror Universe, tribbles, Section 31 being a secret organization that no one was aware of, etc.
Definitely agree about Section 31. In both DS9 and Enterprise Section 31 is a top secret organization that almost nobody in Starfleet outside some upper echelon officers is even aware of. In Discovery way too many people are aware of them. Section 31 was designed to operate in the shadows. Being out in the open with starbases and ships of their own completely blows that out of the water. How do you do the things Section 31 does when everyone knows about you. Christ, even some Klingons know about them. Some secret.
Spore drive was a big bugaboo for me as well. I know there were issues with it. But it's a drive system that can instantaneously transport you anywhere in the galaxy (maybe further). Are you telling me Starfleet would just up and abandon it and not try to perfect it or make it safe? Not to mention I can't believe the Klingons and later the Romulans wouldn't do everything they could to steal the secrets of Spore drive (and they likely wouldn't have any ethical compunctions about using it). Hell, if I were the Klingons and Romulans I'd have entire sections of my military devoted to that very goal, stealing the secret of Spore drive. Now, granted, at the end of Season 2 they tried to cover their tracks a little bit by making it all top secret. But I just have a hard time buying with all the people that knew about it that it just disappeared, literally never to be heard of again.
Yes, some Star Trek tech has come and gone (levitating shoes anyone)--but those are once and done plot devices. You never see or hear form them again. Spore drive was a huge part of Discovery's back story. You can't just poo-poo it away as a one time thing.
All that being said, it's one primary reason I wish Discovery was set in the future, post-Nemesis some time. Then none of that would be an issue. They wouldn't need to explain anything because it would all be future technology--and maybe by that later date Section 31 was more common knowledge (perhaps because of something Dr. Bashir had done since he was trying to expose them). But as someone else noted, by setting it about 10 years pre-original series, they really boxed themselves in with a lot of things, esp. since they stated this was in the prime-timeline, and not some alternate timeline or reboot.
While there was plenty I liked about Discovery, those were 2 things that really bothered me about it, probably more than anything else. And I do think that's story based, not set design based (I have some bugaboos about production design too--but that's sort of a separate issue).