I live in the UK where roundabouts are almost everywhere. While they are definitely more efficient than crossroads they still have some flow control issues in some scenarios: each joining road effectively controls the flow of the next joining road further along the roundabout through priority, this is both a strength (automatic flow control) and a weakness (unbalanced flow of joining roads): Imagine a simple roundabout in place of an crossroad with 4 joining roads, if only two joining roads next to each other are active and of those two, the one to the right has an endless contiguous flow, then all traffic on the road to the left is blocked - it's basically waiting for road to the right to be blocked by someone coming onto the roundabout from the next road along.

As a result we have traffic lights on roundabouts as well sometimes, which prevents this unfairness but also takes them back closer to the efficiency of a crossroad.

I think the best combinations would be autonomous + roundabouts, I think fast crossroads are just dangerous (autonomous or not). Roundabouts force vehicles to slow down due to the forced turn and negotiate one stream of traffic going in one direction rather than two streams going in up to 4 directions at speed. The potential injury from the later is so much higher it's a non-argument.

I also live in the UK, so I'm familiar with roundabouts for the same reasons you are, but I'm glad you've given the explanation for those that are not.

In your example, "an crossroad with 4 joining roads, if only two joining roads next to each other are active and of those two, the one to the right has an endless contiguous flow, then all traffic on the road to the left is blocked", this is a theoretical weakness, but in practice it rarely occurs. For starters, the chances of zero traffic from the other two roads is small, and "continuous" traffic is rarely busy to the level where cars from the road to the left are blocked completely.

Furthermore, if such bottlenecks occur, they can be mitigated against by diverting traffic, through the addition of changes to surrounding roads.

Roundabouts may not be something to be implemented without consideration to the surrounding road infrastructure, but they seem like they would be decent step forward for many busy intersections with traffic coming from multiple directions.