My local MP has been banging on about this for some timeand indeed commented on it today. They've been interested in SE metro devolution for some time; whether anyone is prepared to actually give it a detailed look to try and split off the Metro side in a manner that works and boost frequency on a congested route is a different matter. If TfL did run it (highly unlikely due to finances etc.), passengers will probably need to be prepared for some unpopular decisions regarding rationalisation of London terminals.
All the London terminal or key sections on the approaches are at capacity so there won't be any higher frequencies unless the Hayes line gets Bakerloo extensioned and those 6 paths / hours get reallocated to the other 4 via LBG metro routes.
Hence capacity increase is effectively via longer trains and changes to seating arrangements. But depot space and stabling are at capacity so this would need £200-250m minimum to address on top of any additional rolling stock.
All the previous TfL plans largely focused on improving reliability though simplification (services from stations down to 1 / 2 termini from 2 / 3 termini currently). This won't go down well with many users and Lewisham won't be able to cope with the increased interchange volumes without spend at least £70m on it. The previous TfL thinking also didn't include the then future additional Thameslink paths post London Bridge completion so many of the opportunities aren't do able in practice.
A lot of the TfL South London Metro (not just) thinking involves DfT/HMT giving NR about £1.5bn as starting point for infrastructure which they conveniently leave out of "their" costings which are effectively done on a TOC basis.
At this point is is probably worth waiting till crossrail phase 3 (Abbey Wood - Paddington) opens and the Thameslink Maidstone/ Ashford service starts up before assessing the effect on the SE network. A sensible direct award with more gating and staffing to capture actual user numbers would be a sensible to make accurate assumptions.
12car via London Bridge SE metro is much higher on DfT list than TfL's...