The problem is that (higher) exponential growth will cause many more hospitalizations to happen in a much shorter time interval. That can make the wave more deadly even if the virus is somewhat milder.

You're missing the point. If Delta kills, say, a million people, and Omicron only kills half a million people, but Omicron kills that halved amount in, say, 1/10th the time, Omicron is more likely to lead to overwhelmed hospitals.

I wish they would just triage unvaccinated people below everyone else so hospital capacity would stop being a problem.