I live in the world in which this is how these things work. In my neck o' the woods, when your car winds up in the impound lot through no fault of your own -- i.e., in a situation exactly as you describe it -- there are procedures in place to ensure that you don't have to pay for these things. Now, is it all rainbows and unicorns? No. You gotta fill out these forms in triplicate and do all sorts of affidavits and the like. And you gotta deal with the bureaucracy and all that that implies.
Apparently you'd set it up such that if someone's car winds up in the impound lot, all he or she has to do is say, "Hey, it was stolen!" and the gates would just magically open and the people who paid for the towing (and maintain the impound lot) would say, "Hey, this one's on us!"
But I have it in my head that your little dream world would turn out to be a nightmare ...