If it was the case, I would say its much easier for one terminal to be the "Overground" side rather than the other. At present Charing Cross is more "mainline" than Cannon Street (outside peak specials, and who knows how they will go in future), so Cannon St is the more logical option. It would also allow the semi-fast to continue to run to Gravesend from Charing Cross as a Southeastern service, if that was wanted.
The easiest way to do things like metroisation (which I posted upthread is a bit of a pipe dream anyway right now) is standardisation which the Overground (and TfL Rail) love and it does make sense for that sort of service. The problem is passengers are actually very happy with their rather messy and irrational "all stations to all terminals" service which no doubt the Overground would chop up and indeed there would be much more changing at London Bridge. So yes the Overground would offer benefits, but there would be a bit of "give and take" as well.
I think if a better service is available, passengers will survive and adapt. Chiltern passengers don't have much choice in terminal at all.
London Bridge was rebuilt for this exact purpose. Segregating the terminal lines and massively improving the interchange experience. Lewisham is subpar for this of course, but is an option for certain links.
Not to mention, there is no sole terminus. They all get London Bridge and everything that offers too - connectivity but a major destination for jobs and leisure too. So it's a bit silly.
Crossrail will help a good chunk of them in variety of locations, as HS1 did at the eastern edges for Medway - and the DLR and ELL have added new connectivity to SE routes over the years too. London has evolved eastwards too, and is less concentric in commuting. So I think those grumbles are slightly archaic and less relevant to the polycentric nature of today. That argument is over. The SE network 20 years ago was very single purpose. The map is very different now, as are the demand centres.
Make the change, people will adapt. And if their services are all 6tph and evenly spaced, as good as an outer tube line - they'll soon forget.