The problem with 'a charity covering the bill' is that you then need people to donate out of goodwill. That doesn't often happen.
There's already fines and punishment for abusing emergency support systems (e.g.: calling 000/911/your country's equivalent). Even if you abuse that, and even if you mislead the call centre operator to dispatch an ambulance to your house, they're not going to ferry you to your desired location unless your vitals show reason to consider it a possible requirement.
Ambulance services are free here, insofar as you pay for a small tax on every rates bill (said rates cover ambulance and waste management services) edit: Apparently not since 2003 - it's now simply just covered by the State. There's no out-of-pocket expense. I can guarantee you that we don't have ambulances running people not requiring their services around 24/7.