Not to be too snarky, but EMTs don't carry morphine. Restricted substances like morphine are only carried by paramedics (or some AEMTs in systems that still have them). As such, paramedics are dispatched to calls that require a higher level of care while ambulances staffed solely with EMTs are sent to less emergent situations. This lines up pretty well with what the GP described.

This is in reference to American standards of care, other countries may be different.