I'm an American who lived in the UK for a couple of years. They have many roundabouts there, and I love them. But now I live in an American town with a few, and I've realized that there's a reason that they are less safe in the US than in the UK--besides the fact that American drivers are less familiar with them. It's a solvable problem, but not one I've heard people talk about. (Though to be clear, I'm not a traffic engineer.)
Most American drivers are familiar with slip lanes, which allow drivers to make a right-hand turns without necessarily coming to a stop. (Wikipedia article: https://en.wikipedia.org/wiki/Slip_lane. Note that the diagram shows an example from a country where they drive on the left side of the road.) They're convenient for drivers, but dangerous for anyone on foot or a bike. This is because a driver in a slip lane is looking for an opening in traffic in the direction opposite of the direction they're traveling.
A roundabout is basically an intersection made up of nothing but slip lanes. So they're fundamentally dangerous to pedestrians in the same way that a slip lane is, but the fact that vehicles are moving slower means that they're still safer than typical American intersections.
However, if I remember correctly (it's been over 20 years ago), in the UK they don't mix roundabouts and crosswalks. They'd put crosswalks (called "Zebra crossings": https://en.wikipedia.org/wiki/Zebra_crossing) between intersections, where drivers aren't distracted by other things. I think we need to do that in the US as well if we're going to adopt roundabouts.