Brian, yes Sydenham does have a bridge, but you can get off at the down platform and walk round to the other via the street. This makes it possible. However it does take time and I would rather not have to do it. Also, just as people do not like the FH underpass, the street route to the up platform is not well lit and also a little scary after dark- which until recently was 4.30pm -5pmish. Just to break the monotony I travelled from London Bridge to Catford Bridge yesterday rather than have to travel via Sydenham; it took me an extra 50 minutes yes 50 as had to wait a long time for a 185 bus which did not already have its full quota of buggies aboard, and then there was a lot of traffic heading back up towards FH so the journey was slow.
In a world where there are so much emphasis on increasing accessibility for all, its a shame that things seemingly have to get so much worse for people before they get better.