It's not possible to walk through somewhere like Lace Market tramstop without either walking on the platforms or walking on the road, and their revenue blocks have to cope with that so could presumably do the same at the station tramstop.
At the south end it's easy enough to descend from the tram platforms into the station building and out at street level underneath the viaduct, but if this was a public right of way it would have to remain open 24/7 and they'd probably need another lockable barrier to prevent people wandering into the rest of the station while closed. It lands people on a fairly unpleasant bit of Queens Road with no pedestrian crossing for onward travel, but then again so does the existing right of way.
Putting the right of way on the tram bridge would still mean closure of the station access from Station Street and probably also the one onto Platform 7 from Queens Road, so access would only be possible via the Carrington Street "main entrance", through the car park or from Queens Road under the viaduct. In theory it would also be possible to enter the station via the Station Street steps and the tramstop, but anyone doing that would be quicker walking round to the main entrance.