basically everything was plot, which made the emotional parts feel unearned.

Unearned, and also shoehorned in. They were in the middle of a series of crises, and instead of just putting the personal stuff to the side until the crisis/crises were over they had to deal with personal soap-opera stuff in the middle of that. And, that meant that you couldn’t have personal character development that was low-stakes. For it to interrupt the crisis it had to be high stakes. That just heightened the soap-operaness because every emotional moment was high stakes.

Discovery lacks that kind of deferring to better expertise, and often comes across as Burnham does everything.

That’s basically what I mean about the incompetence. She had to do everything herself rather than consult with the rest of the crew, often breaking the rules because she didn’t have time to follow them because everything was so urgent. On every other Star Trek, the chief engineer would be consulted when it came to engineering things, the science officer when it came to science things, and so-on. But, because Burnham didn’t consult her experts, it makes it seem like they’re not competent enough to keep up with her.

So, these other crew members are involved when there’s a high-stakes soap opera scene where they bare their souls. But, they’re bypassed when Burnham has to take quick actions or the whole multiverse dies. Which makes it seem like this isn’t a crew of a captain, a science officer and a chief engineer working together to solve things. It’s a soap opera involving Tilly, Stamets, and Jett while Burnham saves the multiverse.