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In the UK, you get roundabouts with traffic lights at the entrances, sometimes only operational at peak times, and off most of the time. Works well

You do, but that means there is no roundabout when you need it the most.

The problem occurs mostly when the dominant flow is given multiple lanes.

A fairly common solution/workaround is blocking the view of the approaching traffic, forcing it to slow down. But again, this doesn't work well on large roundabouts that allow people to speed up to of 30+ mph.

This really is the big failure mode of roundabouts, it essentially becomes "live-locked"

I used to go to work past one like this. They replaced a light where you would always eventually get to go with a roundabout where the 99% of traffic going north/south meant the other two ways were effectively not part of the traffic network for a few hours a day.