There's no reason why a service road can't be a through route. The problem with track is that it is used for access to land, with the implication that you might need a tractor / 4x4. highway=service + service=driveway might be a good option.

I don't see a problem with the previous "access=private; foot=designated". Changing this to no/yes doesn't change the access rights. I would reserve access=no for cases where the road is impasssable / in a firing range etc. 

If you want to emphasise that the road is open to pedestrians (as opposed to highlighting that it is closed to vehicles), then you could change
access=no/private 
to
motor_vehicle=private

This means the same thing, but it won't show the markings for access=no/private.
