ummm don't cops take a similar oath?  they help save more lives than EMS does on a regular basis, and that's why people want to become cops, not EMS workers.  I don't see people in need not trusting cops because of a sidearm.  yes, the criminals might not trust the cops, but the majority of the public will still call for a cop when they need a cop's assistance.
so you will carry a gun on you when you go to the mall, go shopping, and hang out with the family, but don't think you should have a gun on you with you when you respond to a housing project full of crackheads and drug deals?  or a report of a person stabbed or shot?  or even better, when you respond to a cardiac arrest, and the family says "if you don't save momma's life, we are going to kill you?"
couldn't the same argument be give for a law enforcement officer in the back of an ambulance with a psych patient?  or one who is assisting you in dealing with a hypoglycemic combative patient?