Roundabouts stop being useful once you get to "gridlock" style traffic, though. Because the traffic tends to work its way back into the roundabout and clog the whole thing up. I experienced that this spring in Scotland on a visit.
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.
In Sweden we have a lot of roundabouts (and we also love to brag about having the world's lowest road fatality rate) -- what I see a lot for big roundabouts is traffic lights before entry. Best of both worlds.
I live in a city called Al Ain in the UAE. It's quite known for the sheer number of roundabouts. Due to heavy congestion at some of them, they've started replacing them with either intersections or roundabouts with lights.
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.
That's the question I had. I had a very good experience with roundabouts driving around France for two weeks, but as a resident of Manhattan I wondered how they would fare under extremely congested conditions with many aggressive drivers. (I imagine they have those conditions in Paris -- but in Paris we took the Metro.)
In Paris, the most traffic-heavy roundabout (place de l'Étoile, around the Arc the Triomphe) has different rules from most others, cars on the roundabout must yield to those entering.
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.
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.