I've...never seen a roundabout with lights. Granted, the state I live in doesn't do them much, but I've visited places (like MA) that do them a lot, and they're all very much like European roundabouts, just with inferior signage.
Here's a roundabout in Cape Town (Adderley Street Fountain) with a traffic light - https://www.google.com/maps/@-33.9202591,18.4250697,3a,75y,4... .
Here's one in Cambridge, UK - https://www.google.com/maps/@52.2084812,0.136931,3a,75y,82h,... .
there are a load there with lights, tram crossing it straight, pedestrian crossing that aren't on a road crossing etc. makes for some fun time crossing one at peak hour, it's not weird at that time to get stuck in one for 10 minutes or more.
Some have through streets with lights mixed in with the circle: https://ggwash.org/images/posts/201611-221233-1.png.
Traffic lights (signals), can help improve flow further in these situations as well as giving pedestrians a safe way to cross.
Part-time signals (only on a peak) can be very effective.