911 also decides where the call should go. The problem is that decision often includes the police for liability (sounds like the person might be violent, even if the root cause is mental health) and/or logistics (a fire usually requires traffic/crowd control) reasons, and they generally respond the quickest (because they don't need to put on their gear, are already patrolling, and have smaller vehicles).

Whoever is initially dispatched to the call by 911 also have a say. E.g. if 911 dispatches police, but the police feel they want an ambulance on scene as well, they can request that. And vice-versa.

There is a difficult balance to strike, in that police presence can escalate tension and make a situation worse, but police are also best equipped to protect other responders if a situation does escalate. It can be a tough judgment call