The rest of the problems almost all come down to lack of setup, just as with the last minute characterization they gave Airiam, the cyborg woman, in a desperate attempt to get the audience to care about her just before killing her.
There's no setup or follow up for why Burnham thinks the giant dilithium world is important to the Burn.
No connection in theme, context, or plot for the Burn source. Unbelievably, Burnham being the Burn source would make more sense.
They have a cloak, which should have been mentioned in episode 6 with the upgrade. They sure mentioned a lot of other stuff which didn't need mentioning. It would have created either an expectation or a forget it and surprise situation, and would have removed Tilly the first officer having to ask if her ship has a system. If anyone knows the ship has a major system, it has to be her.
Apparently nothing, not even shields, stop transporters now, and no one thinks to just beam torpedoes into each others ships.