Various possible reasons. My hypothesis is that the railway always owned the branch from Silverknowes Road to Barnton but disposed of it in stages, and LRC was unable (or unwilling) to develop the remaining section.
Barnton Station, immediately north of the Barnton Park Hotel, was the first to close in the 1950s I think, and when that happened the wayleave west of Cramond Road South became redundant. The western half became the houses on the north side of Barnton Park View which was then just starting to be built. Later on, the eastern part became nos.1 to 11 Barnton Avenue when the Barntongate Nursery was sold for housing; and the middle section became the driveway to the Royal High School.
But the railway from Silverknowes Road to Cramond Road South wasn't all lifted; it was singled because the goods station was still in use. The goods station closed in the 1970s and the Safeway was built there (now, Tesco Metro). I have a vague feeling that it was part of the planning application that the supermarket carpark would not provide official continuation of the cycle path. In the mid-90s that path was still just a muddy track. But by then the alternative route – what we all know as route 1/76 – was already established.
It is also possible that LRC chose not to develop the supermarket section because the exit onto Cramond Road South was too close to the main junction, and Cars Must Not Be Held Up By Pesky Cyclists Who Just Put Themselves In Danger Anyway.
That stupid chicane gate onto Cramond Road South is therefore a constant reminder of what could have been. Heck, what could have been was a dead straight cycle path that went all the way to Whitehouse Road. Heck, what could have been was a railway, or tram, that served the northwestern part of Edinburgh.