I think you need to spend a bit more time in the UK !
Take Hyde Park Corner for example. A bit of an extreme example as it is a super-busy roundabout. I will grant you that its not one where you will make smooth progress during peak rush-hour (outside of peak it works perfectly though).
There are also other good (and less extreme) examples along the secondary routes out of Heathrow (i.e. along Bath Road and Great West Road).
There's no way in hell you could replace any of the above with a US-style multi-way stop and achieve the same throughput of traffic.
Roundabouts work so well because you can have that magical combination of vehicles already established on the roundabout and also adding new vehicles all the time.
If you try to join via the wrong lane then you're in for a world of hell. ;-)
I don't find this to be accurate--at least in the US and in my limited experience in the UK. In computer terms, roundabouts don't guarantee progress under contention.
Roundabouts seem to work when there is semi-continuous, medium-level amounts of traffic. They are better than stop signs, better than unsensored signals, and probably comparable to sensored traffic signals (which are more expensive).
However, once the traffic becomes high, roundabouts are TERRIBLE. Stop signs and controlled intersections at least guarantee some progress while someone may sit at a roundabout entrance almost indefinitely since "give way to traffic on the roundabout" is the normal rule. This is particularly bad when you are entering a roundabout between the most popular inlet and the most popular outlet.