Before V1 you need to be able to abort the takeoff and stop safely in the remaining runway. After rotation the task is fundamentally different, so V1 cannot be higher than Vr. If you don't plan to rotate at "Vr" it is not Vr.
I was thinking about when V1 might apply in a single. There are problems where the correct thing to do is continue the takeoff so you could say V1 does apply. For example, if a door pops open in the takeoff roll.
Early in the takeoff roll, if a door pops open, cut the power and come to a stop, no problem.
If the door pops open as you rotate, cutting the power and trying to stop on the runway is a recipe for a broken aeroplane. Better to continue the takeoff, and do a normal circuit and landing. Or if the runway is really long, maybe you can stabilize the aircraft, reduce power and land straight ahead. But again that is different to an aborted takeoff before rotation.
If the runway is short, e.g. you have a 500m runway and calculated a 400m takeoff distance, at some point before rotation you are committed - you probably can't stop in the remaining runway. At that point, if the door pops open you need to continue the takeoff. That is your V1 - even if no-one ever calculates it.
There is no reason for V1 to exist except "what it is for". If the definition doesn't quite capture it there is a problem with the definition. "What it is for" and how it is used are the whole basis of V1.
V1 is most critical when the runway length is limited. Yes. on longer runways V1 can be equal to Vr and you can abort the takeoff any time before rotation.