As a manual wheelchair user I'm sure I could get through there without problems, I'm not sure about larger powerchairs and mobility scooters though...
The graphic is very good, barriers tend to be used in an attempt to limit illegal use, often by motorcycles but unfortunately they just tend to stop legal use by people like me ):
Areas I avoid when handcycling around Edinburgh are the speed bumps on NCN 1 at Balbirnie Place (as I ground on them) and the footbridge on NCN 1 over the river Esk near Monktonhall as the drop off going east is bigger than any kerb!
There are also the (probably illegal) barriers at each end of NCN 76 at both the Dalmeny and Hopetoun estates. The barriers at Hopetoun at the quarry gate and Deer Park road are especially irritating as I used to regularly handcycle through the estate this way before the route was changed and the ridiculous gates were installed around 15(?) years ago.
There are lots of chicanes that are passable with difficulty like those either end of the Goose Green footbridge in Musselburgh and the bridge over the Newbridge roundabout. I haven't tried to go west along the canal for many years after finding several barriers between the city and Broxburn but I understand the gates along the canal have been opened and it might now be passable all the way to Falkirk.
Things have improved no end in Edinburgh over the past 20 years though I now tend to stick to roads.