The potential for using Gifford Park and Rankeillor Street has been mentioned on another thread.
At present it's blocked by a railing and hampered by the wheelie bins (check area on Google StreetView).
It really shouldn't be difficult to get the railings removed (but This Is Edinburgh). Obviously some measures will need to be taken to reduce cycle/pedestrian conflict. This would probably involve the nasty chicane barriers that cyclists hate - but they do a job.
Cycle routes that cross pavements already exist in Edinburgh - e.g. Raeburn Place (no barriers but much better sight lines).
The pavement in Clerk Street is not particularly wide so a build-out where people on bikes would wait for a gap in the traffic would be a good idea. It would also make it possible for some of the wheelie bins to be positioned to the north of this.
This would be in the current bus lane, which would have to be moved further out into the carriageway. Clerk Street is significantly wider than South Bridge so that needn't be a problem...
Obviously making all the pavements in Clerk Street wider would re-balance the space between pedestrians and vehicles and probably re-vitalise the 'local shopping experience' - but This Is Edinburgh...
At present there is a pedestrian crossing to the south of Gifford Park/Rankeillor Street. This stops the traffic periodically. Technically the lights could also be triggered by bicycle detectors in Gifford Park and Rankeillor Street.
Perhaps CEC could innovate with a wireless connection between the detectors and the crossing controls(???)
What would be even more adventurous (and be of use to pedestrians, bus users and cyclists) would be an additional crossing to the north of Gifford Park/Rankeillor Street which would operate in unison with the existing crossing.
This would have almost no effect on vehicle traffic flow (which is what tends to concern planners/engineers/bus companies) but a positive effect on cyclists using the Gifford Park/Rankeillor Street route.
A 'final' improvement would be to make Rankeillor Street one way to motor vehicles (west to east) further improving the crossing for cyclists.
After some or all of this is done, attention could be paid to Buccleuch Street and St. Leonards Street.
If/when the site at the end of Lochrin Place is built on there will be (?) a new link to the Canal which would avoid the Gilmore Place/Kings junction (though re-doing the signal phases here would improve things for cyclists).
SO one day there might be a decent cycle route (mostly off-road) across the centre of Edinburgh that would attract all ages and most abilities.