We fly into a private site fairly regularly and have placed an old car there for several years now. Only issue is MOT every year which is a pain to organise at a distance.
We recently had someone do a similar thing at our home airfield to visit a sick relative.
Insurance would be the main worry of allowing a general use car. It's a good idea but making it work could turn into a burden.
I think it would actually be easier at smaller strips, where there would be less demand and pressure on its use.