I'm not sure that that is completely true. While in the UK, I would have had a lot of difficulty registering with a GP in a whole other city.
Well, depends on how close the city is. Here, I grew up in a small town that didn't have any doctors. I had to drive 30 minutes to get to the larger city that had doctros' offices and hospitals. But even within the same city, some doctors are part of one insurance network, others are part of a second, still others are part of a third, and on and on. There are quite a few that are "in-network" for multiple networks, but it's quite a mess to try and make sure your doctor is aprt of the same network you are.
For instance, my wife and I are hoping to have a baby. My wife knows what hospital she wants to deliver at, and it's part of the right network. But even so, we need to make sure the doctor she wants is part of the network as well, despite the fact that the doctor works in that hospital.
With public healthcare, we could just go to the hospital we want, see the doctor we want, and not have to worry about what plan s/he is in and whether ours is the same.
As for a GP, yeah, you'd probably need to register and make sure the doctor has the capacity to add another patient, etc. But even so, you'd have one less worry, that of making sure the GP is part of your network.