The problem with an "East Coast" route is you have nowhere of any significance between London and Leeds. The best capacity increase on the East Coast could probably be achieve by eliminating 2/3 track sections like Welwyn and Connington - Peterborough.
I would think that that the interventions other than Welwyn which would be most useful would be
1) Rebuilding Newark Station with through roads in association with grade separating the flat crossing would be the most useful intervention. As well as removing the flat crossing you provide a station layout that allows fast trains to overtake those calling at Newark.
2) Second priority (perhaps it should be first as much cheaper than (1). Providing "goods line" platforms at Hornsey and Haringey so that Hertford Loop services stay off the down slows which in term means some GN/TL traffic has less fast line occupation (as well as being able to provide a far more frequent Moorgate to Hertford service).
3) I would make third priority on an additional down platform at Retford so the existing down fast platform could be closed off and Retford had two through roads for fast trains to be able to overtake semifasts calling at Retford.
Ideal but probably wouldn't stack up in a business case:
4) Fourth, complete Huntingdon to Peterborough re? Quadrupling.
5) Extend the 4 track to Grantham with a new bore for Stoke Tunnel suitable for high speed passage. I would also rebuild grantham to have tbrough road.
Kings Cross is a fixed constrain, however it has nine intercity platforms for less intercity trains per hour than St Pancras has with four platforms on the Midland Main line next door. Therefore replanning of how trains are handled as was done at St Pancras when they lost several platforms to Eurostar is needed with long layovers despatched to Hornsey in the same way that they are to Cricklewood on the Midland Main Line.
A good start would be to end the "temporary" situation of the two Thameslink stoppers to Cambridge terminating at KX platform 3 instead of going down the Thameslink core.