It isn't the mushroom drive that made Discovery bad, it's that Starfleet apparently no longer has any kind of standards.
TOS and TNG had all kinds of "woke" politics for their era, but they portrayed them as happening on a military vessel. People were calm, competent and followed the chain of command. The only time that broke down is when they were under the influence of some kind of alien disease or tech.
Discovery's crew was full of whiny, fragile people that were barely able to do their jobs for all the time they spent obsessing about their personal problems. Tilly is the prime example of this. The "Tilly" equivalent in TNG was Reginald Barclay. Shy, stressed, lacking self confidence, etc. Barclay's character arc makes sense for Star Trek. He is able to save the day, but he's certainly not promoted because it's clear that the senior officers on the show are calm, competent and project confidence. He's basically there to show that not all Star Trek characters are the confident, competent, brave people who make up the bridge crew. And, by doing that they emphasize how elite the bridge crew is. Meanwhile, on Discovery, Tilly is promoted and keeps gaining responsibility despite never addressing these gaping character flaws. The "Tilly" message seems to be something like "it doesn't matter if you're weird, awkward and unable to communicate competently, as long as you love and accept yourself, you too deserve to be on the bridge making life or death decisions".
Discovery also fails because that lack of competence is everywhere in the crew. The original shows had the crew acting as... well a crew. They'd tackle problems together. In TOS Kirk would lead the charge, but he'd never do anything on his own. Spock was stronger and smarter than anybody else, but he followed the lead of his commander. McCoy handled the medical stuff. Scotty handled engineering. In Discovery, Burnham is apparently the only competent person on the crew, and the only one not to be fazed when something bad happens, so rather than the crew working together to solve issues, it's superhero Burnham while the crew faints dramatically. The only real exceptions to that are Saru (whose personality doesn't really make sense given what they explain about his species), and Commander Reno, who is a breath of fresh air because she's basically the only one who isn't constantly freaking out -- although the sarcasm and fatalism of her character is almost too much.
What makes it all worse is that the backdrop is that the universe is doomed and only Discovery can save it. Sure, the other Treks have had major threats to the universe, but they were being slow-rolled over a long season, or sometimes multiple seasons. They had room to breathe and do episodes that didn't advance the plot. That gave them a time to do episodes focused on fleshing out the personality of a member of the crew, to do silly things, etc. Discovery has the whiniest, least professional crew that has ever crewed a starship (and I'm including Boimler and friends), who are whining while dealing with the most urgent apocalyptic scenarios. It's a soap opera while the end of the world is playing out.