You're right that the Thames Path isn't a Public Right of Way, like many tracks across fields in the countryside (Inner London was specifically excluded from the PROW legislation, so it couldn't be). Shared spaces (like Exhibition Road or, more locally, Baron's Place) don't fit my suggested heuristic, but they are very much a minority of streets.
Access on bicycle is, as you say sporadically expressly banned. The signs at either end of Gabriel's Wharf are about as unclear as the ones at More London, and even less respected.
I can't comment on the original point, as I'm not familiar with it, but if you want to avoid getting on and off your bike, there's a useful view of the OS 1:50,000 for the South Bank clearly showing the Thames (foot) Path - red diamonds - and the parallel Thames Cycle Path - green dots - here. You need to zoom out twice to get the route markers.