So long as the traffic flow is at least the double-digits (mph or kph) roundabouts are just fine. Below that though, and there be dragons.
People tend not to jam up intersections with stop signs unless there's a bit of space for them on the other side of the intersection. Same for stoplights, although there are some cases where traffic backs up into the intersection though that generally has more to do with people who are moving (albeit slowly) and then everyone jams on the brakes all at once, stranding several in the intersection.
How does congestion happen? A roundabout typically has four entrances. During peak traffic, one of these entrances might get clogged. This one side causes the other four sides delays due to right of way. Say a buildup forms at another side because of the delay. The other sides will be even more affected. This snowball effect is usually what causes congestion at peak traffic.
Traffic intersections on the other hand are immune to this issue because the delay depends solely on the light's timing, making it way more predictable.
I'm not sure it improves safety, but I think it helps preventing gridlock.
My experience in Nantes, a city with a lot of roundabouts (know for double roundabouts[1] and square roundabouts[2]) is that double roundabouts seem a bit better when there is congestion.
Usually when there is congestion on a roundabout, people on the roundabout will let every other waiting entering car pass before them, anyway. I think it works quite well.
[1] https://www.google.fr/maps/place/Rond-Point+de+Rennes,+44300...
[2] https://www.google.fr/maps/place/Place+du+G%C3%A9n%C3%A9ral+...
It gets fun when you have the tram line going through 1
Drivers who are comfortable, having fun, and moving continuously are much more likely to speed and get in accidents than drivers who are annoyed and constantly vigilant for conditions changing second-to-second (like stop-and-go traffic).