It doesn't look like it was ever a passenger bridge, from this
1949 aerial photo in the Canmore collection it looks like it had lift winding houses on it, so it was possibly for transferring mail or parcels, or for taking catering supplies to trains (the central of the 3 bridges has always been the main passenger footbridge).
This 1971 interior photo from the collection shows it ending above and going down into some kind of office (possibly a parcels office?). By the time
this 1985 photo from the collection was taken, the concourse has been remodelled, the office has gone and the bridge is in it's present state (though partially hidden by an advertising hoarding).
It looks like when Platforms 8 & 9 were taken out of use (the two westernmost through platforms adjacent to College St) when the station was rationalised and remodelled in 1973-74 the northernmost goods lift/bridge was also taken out of use but the southernmost one (which also has a winding house on top of it, but only reaches Platforms 7 & 8) was probably retained to service trains on Platform 7. From
this 1999 photo taken below the northernmost bridge on Platform 7, it seems whatever the winding house was above on Platform 7 was also removed.
The station's through platforms (6-9) originally ran north under Guild Street towards Union Street, and the north-facing bay platforms (10-13) were on the north side of Guild Street, next to the station hotel. There was another footbridge to the north of Guild Street that connected all the platforms to the original suburban booking office on the west end of the Guild Street bridge, which is now a hair salon. The northern end of the through platforms, bay platforms and footbridge were all demolished in 1973 when the station was rationalised, with Atholl House and the Trinity Centre later being built on the site. RailScot
has an article about the demolition.