What about ALS IFT? If you're taking someone from a community hospital to the city that is a mess, is that not an emergency? I bet the sending facility thinks so.
BLS IFT maybe not so much. I don't have to use my toes to count how many times I've diverted to an ED on a transport.
Though the majority of my call volume is not really EMS, we still do 911 backups for the city and occasionally catch calls while covering the 911 town when the medics are out. We have many healthcare (and otherwise) facilities who call us instead of 911 because we get there faster, are nicer, and provide CPR and First Aid classes at a reduced rate.