The walkway on the RBS building(when it was there) was walled in and could (in theory) be cycled safely
The building you are referring to includes the A-listed "The New Club" by Alan Reiach, 1966. I don't think the ex-Royal Bank offices (part to the right, that shares the walkway) are included in the listing
Apparently the interior of the club parts rescued the original interior of the original baroque building, so is completely unlike the modernist exterior.
The idea for the first-floor walkway was part of the 1954 "Princes Street Panel guidelines". There are a few other bits of it. BhS was built to the specification, but with the "walkway" part only forming the overhanging sign, I suppose they might have converted it later on.
I learnt all this from google.
I prefer the following "solution" to Princes Street;
buses, taxis and trams to fight it out on Princes Street. Bi-directional segregated cycle lane to be installed on George Street giving rapid and safe east-west transit through town. Single lane each way can be maintained on George Street for deliveries and taxis etc. Centre parking lane to be removed as surplus to requirements (big empty multi-storeys nearby at Kings Stables Road, Omni Centre and St. James Centre. Or they can take the tram!)
I prefer the George Street idea as more practical given that it's now going to be impossible to re-locate the tram lines. I'm also of the theory that it benefits most people wanting east-west transit through the city centre, anyone wanting to get to the shops can turn off relevant side street (Castle, Hanover, Frederick, etc.) and park up there.