Maybe they should split the Northern into two notionally separate lines,
Northern Bank Line and Northern Charing Cross Line.

That's exactly what TfL want to do.

Quote Wikipedia:

" TfL's Transport 2025 - Transport vision for a growing world [4]
confirms their ambition to split the Northern line into two segregated
routes. Running trains between all combinations of branches, as is
currently done, means only 22-25 trains per hour can run on each of
the central sections, because merging trains have to wait for each
other at the junctions. Segregating the routes will allow 30 trains
per hour on all parts of the system. However, Camden Town tube station
would need to be rebuilt before this could be implemented, as the
current station would not be able to cope with the increased number of
passengers changing trains there. LUL have however had problems
gaining bureaucratic approval of redevelopment plans[5]. "