I don't think many people will argue that running from the police is something we want to legitimize. It is also is never going to end well. The police have orders of magnitude more resources and will eventually catch you. (And meanwhile, you've probably put hundreds of innocent lives at risk while speeding through traffic.) The correct venue there is to argue your case in court.

As for the hospital... that's a nuanced one. There's certainly an argument for that, especially in rural areas where ambulances are not readily available. There's also the argument that without lights and sirens, you're putting everyone else around you at risk. There's a reason the law doesn't carve out an exception for private individuals speeding in medical emergencies.