Hi. ER tech here X 3 long years.
Our hospital is not a teaching hospital, but respectable in town for cardiac/stroke patients...thus bringing lots of wanna-be MD volunteers and workers.
Let me tell you how most EMT/paramedics are perceived here. I've trained many EMTs ...and many of them are unfortunately there for "resume building" and didn't really care about actual learning. Some were plainly dumb, and would be quite scary if they ended up becoming a doc. I know some paramedics who still can't distinguish cardiac death in EKG. But then I've seen medical students asking me if Big-O ST depression meant anything. (God help us) I've seen lots of EMTs/Paramedics/Firefighters who wanted to become a MD. They've came to ER personally to get a rec from MDs. Buying them a drink or two and asking with smile. I've asked that MD if he was going to write favorable rec. He said, "I don't know him that much. I don't know his clinical strength, so all I can say is good luck."
so if you want a good rec from them...show them your strong clinical side.
this is hard to do unless you've worked there for period of time.
As far as your "some valuable networking with physicians," physicians usually see EMTs as "part time staffs," and don't really care to talk to you. ...unless you make some real effort to get to know them personally. I know volunteers who are university premed students, and all they do is transporting a patient to places...don't get to meet MDs this way....
I've got to know some MDs and DOs over the years, and they will likely to write my recs but it took me awhile to be in their good side(hundreds of exams I had to assist).
My recommendation would be to work as EMT for periods of time and get to know people. Or talk to clinical supervisor/ER sup about your volunteering. They can always use your help.