Yes, I agree with the comments in the preceeding posts. Southern simply does not want to run relatively short journey trundlers, preferring to concentrate on longer, presumably higher value ticket routes. I can't see this changing and doubt whether there will be much action under the present government (given that inner London tends to vote the other way). Hence the "you have the Overground, you can like it or lump it".
To be honest I would prefer to see the short Southern train services transferred into TfL as at least there would be one body co-ordinating short journeys into and around London. But we know TfL money woes, and again I don't see this happening in this parliament given the poor relations between national government and London Mayor.
On a more practical basis, I have been to Gatwick Airport a couple of times from Honor Oak Park, my preferred route is journey to London Bridge and head to Platform 5/6 for the Brighton trains there. It annoys me that I have to do it, but only one change and am more confident in this route than doing the go to Norwood Junction, change, got to East Croydon change. And there are lifts at LB. Then again I have a over 60 pass - not sure if it works so well if under 60.