The thing is, the WGH was never served by railways: it didn't even make the transition from poorhouse to hospital until nearly 1930. It is where it is because you built poorhouses in faraway lands to keep one's city's fine upstanding citizens from mingling accidentally with the great unwashed, and Craigleith Station was a sufficiently long walk away. Replacing a railway with a walking and cycling path and replacing that with trams-to-WGH doesn't make the location any closer. Public transport has to serve trip generating locations as physically closely as possible.
The NGH had a small station; while the EGH also found its situation because of its poorhouse history, it too didn't have a rail connection, but did have trams along Seafield Road.