W&C was originally part of the rail network, and for getting people to
the city from routes that came into waterloo. The routes that came into
blackfriars were a rival to the waterloo-w&c option, so I doubt the
people who ran either line would have been too keen on those, in a
similar way to the fact that the circle/metropolitan line doesn't go to
euston (a rival route to the backers of the original line from kings
cross).

It only became part of the tube recently, so I guess they haven't had
much time for that, but maybe they might be planning something like
this when they demolish Blackfriars and move the station to the bridge
instead?