Yet emergency care is relatively small portion of health care costs - so maybe deal with that last. It'd be hard for hospitals to make the case that emergency care is substantially more costly than similar procedures they offer.
There was a thread here a while back about something like an emergency-mode uber for replacing ambulances for most ER trips. Something similar might also be done for shopping for cheaper urgent care too perhaps. Say a clinic that is open late and will check out your scary rash/cough/pain without charging so much.
There will always be a trade-off between quality and cost. If you accept nothing less than highest quality (enforced by law and an extremely litigious society), then you get the US hyper-inflation of healthcare prices.
Also in the US, I think ambulances typically only have EMT's and paramedics. Though sometimes it can cost as much as the ER itself.