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.
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. ;-)
There are hundreds of reasons why. But I'll just name the most obvious one that you've missed.
Traffic volume.
Multi-way stops just kill traffic throughput.
Its a hard fact. You make each vehicle stop.
Meanwhile roundabouts can safely and efficiently move staggering amounts of traffic. Want to move more traffic ? Add an extra lane or two.
P.S. As for the bicycle saftey...
Historically yes, but only to a degree, because I'd argue that if you can't manage to avoid hitting a bicycle on a roundabout then you probably should not be driving because you are most likely also committing lots of other dangerous driving offences elsewhere.
Going forward however, it should be stressed that a lot more attention is being paid to cyclist safety, both in terms of further re-enforcing driver awareness (i.e. more prominent in driver training and examination) but also in terms of vehicle design (especially HGVs) and roundabout design.
I could also argue that with a multi-way stop, perhaps especially at night or in inclement weather or where HGVs are at play, that your cyclist might not be immune to death or injury.