Googling suggests that back in 2011, the council were talking about making it into a right of way (RoW), but didn't take plans forward, because the then SofS Justine Greening came up with £3m for another bridge.
The problem seems to me that many people seem to believe it already is a RoW based purely on what they want it to be; and it just doesn't work like that. I doubt the council could unilaterally make it so anyway, because NR and EMT would show that over a long period of time it has not been continuously open...
It isn't a PROW. Nor likely to be. However the issue is that ....
1. Public money was expended to build the bridge and its entrance to the tram
2. Tram passengers have to cross the bridge to reach the ticket office and station facilities, as well as the bus station. The ticket machines at the tram stop entrance can only offer a fraction of the available tickets from Sheffield.
3. The tram stop is called Sheffield Station and Sheffield Hallam University - there is no convenient way to reach the University other than via the station bridge. Elderley and Disabled people cannot use the horrible alternative bridge which is about 500m further south as both ends are accessed by steep steps.
4. The residents of the recently restored Park Hill Flats can only reach the town via the station bridge.
5. The tram network and the station are both operated by Stagecoach companies.
There is considerable local resistance to barriers. Barriers at the foot of the platform staircases would be both a safety hazard - passengers queuing on staircases, an impediment to changing trains. Similarly barriers on the footbridge would also detain and delay passengers changing trains. There are four staircase entrances from the central island platform (2,3,4 & 5) and two from platform 6,7,8. The entrance /exit at the station front leads into the concourse, but also serves platform 1.
Practically the station does not lend itself to barriers.
On train inspection needs to be beefed up, perhaps with squads of RPIs travelling on Northern trains between Meadowhall and Sheffield. EMT can do similar between Chesterfield (is it gated?) and Sheffield. XC mostly conduct checks between stations anyway.