I suspect the original concept was an answer to a problem that did not really exist in terms of traffic flows. From the North, it was meant to ease congestion at St Pancras, and was particularly useful with the service into Moorgate. From the South, it did not really ease any congestion, mainly because of the junctions described earlier. The cross-London service routes seemed to have been chosen as somewhere to lose the trains (except the Gatwick sweetener), rather than flows demanded by passengers. Even today, I would guess that actual cross-river traffic N-S is a very small part of the overall picture, and that most of the traffic S-N doesn't get beyond Farringdon.
IMHO, the most sensible purpose for a cross-London route would be long-distance passenger (and freight, as there originally was) from the rather isolated South East counties to link with the main network, with usable central London interchanges (unlike the Olympia route).
What condemned the FCC reign to being a dead duck, again IMHO, was dropping Moorgate.