What does randomization mean in this context, and why does it fix those problems?
The same thing it means in every context: that (with enough samples) you can control for confounders.
I think you're correct that randomising patient assignments doesn't control for provider-side confounders. Curious if the study also randomised nursing assignments.
Whether or not they controlled for nurse-alertness is something you'd have to read the paper (or assume the researchers are intelligent) for.
There is also the mechanistic side: if you have lots of plausible mechanism for what's going on, and you can detect indicators for it that don't seem to correlate with nurse alertness, that's a vote against it mattering. Same if you have of lots of expertise on the ground and they can attest that nurse alertness doesn't seem to have an affect. There are lots of ways, basically, to reach pretty good confidence about that, but they might not be as rigorous as randomized assignments can be.