I hope that the pressure regulator would compensate closing/opening valves by keeping the pressure constant but I'm not sure.
In the absence of confounding factors, this is the correct viewpoint.
In order for the pressure regulator to be "perfect", you would ideally have a large volume between the pressure regulator and the valves, and the valves would be spaced a good distance away from each other. In the current picture, the pressure regulator controls the pressure by controlling the flow from the pump. At least this is my understanding.
Non-ideal conditions will be seen when the interacting kinetic forces between the different valves are significant. In that case, the pressure that pressure-regulator sees can't exactly be called the back pressure behind each valve. If we treat the region up to the valve as a stagnant, constant pressure, region then there is no problem with your view. We would also require that the pressure-regulator performs its job perfectly, obviously.
note: I'm assuming the pressure-regulator controls the pressure with its own valve. The representation could be trying to imply that the pressure-regulator is just a sensor and the controller opens and closes one or more of the hose valves, but I don't think this is the system because the OP implies that the hose valves are manually controlled