This is the reason, I am against requiring EMT for firefighter requirements. Please, this is not directly aimed at you but in general. I see several hundred of students a year (Basic to Paramedic levels) so I gather my opinion honestly.
Some FD's are even giving points for Paramedic license here. The problem is most of the applicants could care less about EMS, or health care... and it shows. Again, nothing against those that do, but I find it a rarity and not common. I agree, it would be hard to have an interest and focus on so many multiple hats to wear... and that is another topic.
In to the question in regards, not working on an ambulance affect your registration..that is questionable. Dependent upon your state requirements. Some require that you have to be associated with an EMS (albeit it squad, ambulance, industry,etc) and some do not, as well some require a Medical Director (MD/DO) to attest to your skills. Again, I would check with your local state EMS agency to verify (I have listed all State EMS offices on another link).
For as counting for points, it all depends. Some FD's count the registration as points where others require current registration. Again, one needs to contact their information officer for specifics.
Good luck,
R/r 911