Twin mum here, who will freely admit to having cried, turned around and gone home having driven to a supermarket and been unable to park ANYWHERE I could get the trolley alongside the vehicle (not just P&C spaces, but end of row or near a pillar etc).
That's specifically because our supermarket carpark is on quite a slope and the Sainsburys trolleys here don't have brakes, so it was geuninely impossible for a while to safely leave one child in the trolley and load the other into the car. I will also freely admit that achieving even the simplest things with twin babies was so overwhelming I cried at the daftest things, so that's not necessarily a proportional response!
The P&C spaces are quite intelligently placed though - with pedestrian access right around the edge of the carpark, but at the furthest end of the carpark, right away from the blue badge spaces.
All that happens though is loads of works vans park in the P&C spaces because it's easy to swing into them, and the rest of the carpark is rammed.
I don't think we need a permit system, but I do think the trolley recovery guys could just keep an eye on their use and ask those blatently without children, or people sat IN their cars there to move on.