That would be great, and I'm sure there is demand, but isn't the problem the fact that the network is already chokka with suburban/local services, meaning that there's no opportunity to path fast, non-stop trains? Wasn't that the part of the problem with VXC/AXC services down to Brighton?
Yes, it was. I suppose I'm not saying that it's entirely possible (or likely) but that it would probably be viable in terms of passenger numbers.
I'm going a bit too far into the realms of fantasy here, but BML2 (Uckfield-Lewes) might be able to take some services off the BML, thereby freeing up a bit of capacity. After that, you'd probably want a South<->West chord at Edenbridge. A flyover (or diveunder) at Redhill to carry East<->West services would then be a good plan, with latitudinal platforms. That would allow easier running of cross-country services into Kent via Tonbridge.
Penzance-Dover, anyone?
Sounds nice, but where though?
SWT pulled their Brighton to Reading service, neither FGW or SWT stepped in to replace the XC services from Reading to Portsmouth/ Brighton, same with the old XC service to Kent (Ramsgate?) - there seems to be no TOC interested in running middle/longer distance services in the south east that don't involve Greater London which suggests that there's no market for profitable services there.
You have raised some quite interesting points.
I think the problem is that the TOCs think everyone wants to go via London. This is emphatically not the case. Thameslink and WLL services do help a lot, but I would say the links between the South-East and the South-West are too tenuous.
With more and more people using the railways in the South-East of England, there is probably now a market for direct, fast services to the South-West. Unfortunately, the routes and paths of some of the services before were very slow indeed, which led to a reputation that was hard to shake off. Perhaps destinations such as Ramsgate didn't allow the service to be viable, because a great enough number of population centres was not covered (but that's just my speculation).
I have no idea exactly how to increase paths, but I have listed a couple of infrastructural improvements above.