If you require the police to respond to all medical emergencies, then you have to fund the police at a level which allows them to do that, often at the expense of funding the people who actually should be responding to a medical emergency.

Taking the attitude that the police must respond to all emergency calls even if the only indication of what capacities are needed are fire and/or ambulance services results in a distortion of local funding which makes emergency response worse (it also results in the police presence itself making the response to other things worse in cases where they were not needed, even considering the resources actually available at the time, without considering the effect of response policy on resource allocation -- police are not always neutral-to-beneficial.)

The fact that it is a common rule in the US does not make it right. The US is very bad, compared to other developed countries, at lots of things.