I think it is more to do with the South Eastern railway not wanting to share its terminus with LBSCR. Oops, sorry Southern. Plus ça change...

More seriously, fast forward to London Bridge completion, having untangled the tracks so that the FH service does not impede Thameslink, the planners do not want CX-FH trains to cross in front of current services into CX from Lewisham/Kent etc. So, sadly, I not think we will get CX services back for a generation at least.