Roundabouts are great in many situations but fall down dismally in times of very high traffic volumes, such as peak hour work traffic. If you have bumper to bumper cars all feeding into one large roundabout it can cause a massive tail back as traffic moves into the roundabout at slow, but in bumper to bumper file. Once one feeder direction gains dominance, the other two entry points are simply denied access because no gap opens up. Several intersections near me had the traffic lights removed, only to have to re-install them on the roundabout to avoid the incredible traffic jam.
Many of our roundabouts have traffic lights that only operate during specific times and are switched off otherwise.

There are some new roundabouts in my city. Much crying in the Hammond Facebook groups followed. I can't provide a link, but I have heard they dramatically increase the safety of some busy intersections as well.

People using the road make mistakes (like speeding, running stop signs and red lights, turning left in front of oncoming traffic), always have and always will. Crashes will always be with us, but they need not result in fatalities or serious injury.

Modern roundabouts are the safest form of intersection in the world - the intersection type with the lowest risk of fatal or serious injury crashes - (much more so than comparable signals). Modern roundabouts require a change in speed and alter the geometry of one of the most dangerous parts of the system - intersections. The reduction in speed to about 20 mph and sideswipe geometry mean that, when a crash does happen at a modern roundabout, you might need a tow truck, but rarely an ambulance. Visit the Insurance Institute for Highway Safety or FHWA for modern roundabout FAQs and safety facts.

We've got at least 4 "roundabouts" on Kauai and I love them! They work even when there's a power outage.

The biggest problem is training people in the correct use of lanes on a multi-lane roundabout.
Here in South-Eastern Spain, most Spanish use the outermost lane even when intending to turn 270 degrees.

I lived in DC when younger, traffic circles were common there too.