In the mid 2010s, a TfL takeover of SE looked fairly simple on a map I saw (it may have been posted on this thread so I won’t post it just in case) the map I saw said TfL would take over the rounders and Dartford terminators, leaving the Sidcup line to Gravesend and the Woolwich line to Gillingham with Southeastern.
It was later amended to include the Sidcup route to Gravesend, makes sense since many metro services now terminate at Gravesend to relieve Dartford of the burden.
This still left the Gillingham via Woolwich with SE, again makes sense as this was more in line with the Chiltern service to Aylesbury and the C2C line, a semi fast service running alongside a metro route at a frequency of 2tph.
This was part the reason they shoved Thameslink through this neck of the woods as SE didn’t want this oddity of a service but rather than routing via Blackheath then onto Woolwich and Medway, it was routed via Greenwich, which is primed for a TfL takeover with 6tph at that time, while the Blackheath-Woolwich route is the only service in the metro area that has only 2tph, which isn’t a frequency TfL wants.
Even in new documents the Blackheath-Woolwich route is left as a mainline service to Gillingham