Even the theory is not that clean cut.

What if two cars arrive simultaneously at opposite sides of the 4-way and are both turning left? This could easily deadlock two self-driving cars if not considered.

Your situation is a bit ambiguous. Do you mean they turn to their respective left (in that case their trajectories should not intersect), or to the same left? Not sure about USA, but where I am from the right of the way belongs to the driver who turns to the same side as the side of the road they are on, i.e. to his right (we drive on the right side).

Near where I live we've got two lanes + turn lanes on all 4 sides. It is a nightmare both as a driver and as a pedestrian. I'm not sure how that gets put in place or gets addressed as it inevitably gets busier. Traffic circles perhaps?

Traffic circles work better, but most people in North America have trouble with 2-lane traffic circles. (they're common in most places around the world)

Most people in north america have trouble with 1-lame traffic circles, because they're almost uniformly too small. Traffic engineers like to wedge them in as retrofits, when there's already a lot of stuff built up around an intersection, and it's not feasible to enlarge the intersection to make it work well.

Traffic signals with good sensors are more space efficient, and usually more time efficient when there's congestion, and more pedestrian friendly. Proper sensor design means no or little waiting during low traffic times too. Traffic circles can help with akward intersections though, where there's more than 4 entrances or exits, and some three way intersections work well too.