Paramedics could reasonably be considered as "health care providers" in the common use of the term. However, it's difficult to locate precise confirmation in the FAR.  A working definition of "health care provider" can be found at 29 CFR § 825.125.  While "paramedic" is not expressly listed as an example, paragraph (a)(2) includes "Any other person determined by the Secretary to be capable of providing health care services." Paragraph (c) states "the provider must be authorized to diagnose and treat physical or mental health conditions." Paramedics appear to fit with those criteria. It seems reasonable to include paramedics are "health care providers" even though they are not explicitly listed as such in the Code of Federal Regulations.

It appears from the clause prescription that FAR 52.237-7 would apply, as FAR 37.403 states it shall be inserted in solicitations and contracts for "nonpersonal health care services." To be certain, I recommend you contact one or two contracting officers that awarded the current contracts you refer to and ask the reason for why they did not include it.