Uberuce and I went this way twice today. The piece of shared use pavement under the bridge is completely pointless, as it dumps you out into a road junction about 10m after you get out from under it. There's plenty of road width were to continue the path up Russell Road. Of course there needs to be parking on both sides of the road here...
The viaduct of the WAR is of course a pinch point, with that huge central pier. With the cooncil having their own depot on Russell Road, I'm sure they're unofficially keen on keeping the rat run as it is and not to restrict it to a single lane only under the viaduct.
On google maps I count 5 running lines and 4 lines from the depot to cross on the alignment of the old railway viaduct. That's a lot of railway to cross without some sort of central piers, I can't see Network Rail leaping on the chance to have disruption here, can you?
One option might be to re-model the old embankment from Sauchiebank industrial estate, putting a path up it to cross the spur of the railway somewhere in the vicinity of Duff Street, then run parallel to the WAR, cantilever a bridge out across Gorgie Road from the existing bridge, linking on to existing path behind LIDL. That still leaves Telfer Underpass as bottleneck, but hey-ho...