I imagine that that particular confounder is not possible to eliminate via randomization. Perhaps you collect a bunch of data on nurse awakeness--day shift vs night-shift, measuring alertness somehow, or measuring them on other activities known to be influenced by alertness--and then ensure your results don't correlate with that.
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.
