As someone who is a paramedic, we will absolutely listen to you if you say you want to go to hospital X, not Y.

If you're unable, we'll take you to the one that best is equipped to deal with your issue.

HOWEVER... the insurer. If you're talking about two hospitals in the same town or part of town, no big deal. If you want to go to the hospital 15 minutes beyond that? Your insurer will deny the ambulance bill. Their argument was that "if you were sick enough to require an ambulance, you should have been transported to the closest facility, stabilized, and then transferred to your preferred facility" versus taking a longer ambulance ride.

The principle in that case is elasticity of demand, not competition. As in, my mother always made sure to beat ambulances to us when school or sports reported us injured, because she was worried we wouldn't be confident enough to shoo them away on our own.

how much of healthcare spending is related to emergency procedures?

How much is scheduled weeks in advance? how much is scheduled months in advance.