Which has always seemed a little nuts, like in case of hit and run it would definitely be possible to take action based on a plate alone, both for police and insurance purposes. Unless the registered owner can point to the not-them person who was driving their car at that time, then it was them. Or it was stolen, but either way there will be a clear paper trail.
All that creates a mountain of work and man hours that any police department in America would likely put on low priority.
Basically our legal system is too forgiving and the possibility that someone stole their car (even if it was a friend) and returned it 20 minutes later exists, and therefore it's on the police to prove it wasn't.
And the law is pretty hard to change since it would change it to 'guilty until proven innocent'.
How is it more mountain-of-workish for the policemen if that’s not a rental but my own?
Not true, at least in the US. You are innocent unless it can be positively demonstrated that you were the one driving.
For a serious enough incident the police will invest the necessary time to collect evidence that you were in fact the one driving. But that's costly to do.
I feel the same way about close calls on the road, especially ones involving a vehicle and a vulnerable road user like a pedestrian or cyclist. Way too many lives being saved by a person jumping out of the way at the last minute who shouldn't have had to do so, and then cops and bureaucrats shrugging with "well what do you want us to do, the numbers don't show enough fatalities here for it to be worth fixing" and later when someone actually does die it becomes "this is a horrible tragedy that no one could have seen coming, let's focus on thoughts and prayers rather than accountability that could lead to structural change."