Rounabouts are far, far safer in that they minimize the chance of head-on’s or T-bone crashes. You will get fender benders in a roundabout, but not the horrible crashes we see here.
I lived in Europe for four years and got used to them. They are efficient and safer if you stay alert.
Mythbusters did a test on their show on roundabouts vs. stop signs. Roundabouts were more efficient.
I suspect a lot of Americans don’t like roundabouts simply because they aren’t familiar with them. I have two of them within a few miles of my home, and drive through a third one at least once every week or two. They are fine once they become commonplace enough for people to encounter them frequently.