So spending a load more money on a cross-London link whose primary users* will be London & SE passengers (eg Kent to Heathrow changing at OOC, SW London to Canary Wharf changing at Waterloo/South Bank) will be a really dumb idea to try and sell the scheme to the North...Considering the enormous opposition to HS2 on the basis that it will only benefit London
And that's without dealing with the issue that international journeys on HS1 would still go to St Pancras, which is the other side of London from this new place where northerners end up, but rather having some-of-the-north to Paris trains that won't save much off the journey (and none at all if TGV Picardie is built) vs changing between limited-stop more-frequent services in London due to having to stop everywhere to maximise journey opportunities.
Far far better to build a quality Euston-St Pancras walkway (with travelators and the like) for a tenth of the price of the supposed £3bn additional cost of this (ignoring the sunk costs of the mass of construction that's already happened at Euston).
*when it was direct between OOC and Stratford International, it was modelled as being about 90% of those using the avoid-London route having one end of their journey in Greater London or the SE, and about a third having both ends in it. Add in a South Bank station, and you'd gain some short hoppers doing OOC-South Bank or South Bank-Canary Wharf via the line. Even more London/SE bias!