Gatwick, Luton and Stansted, by virtue of their budget carriers, have an impressive portfolio of European destinations, so they function quite well as EuroAeroHubs. In this respect they ain't broke so why fix them?
The problem is getting to them from any other UK airport, as mentioned up-thread a high speed loop would ease that problem: so you sit on a train for 25 minutes instead of kicking your heels round a departure lounge.
Heathrow, a good solution 70 years ago, continually modified - bodged - ever since, so now there are terminals and associated railway stations all over the shop, however, the lines are only metro and suburban.
Contrast that to Paris CDG, Schipol and Zurich, which have in addition, through European network rail services.
If the proposed new hub, Shivering Sands or elsewhere, was primarily an International Hub it would have 3 categories of arrival passengers; "international" who just change planes and go on their way, "European" who transfer to the EuroAeroHub, "UK domestic" who hop on a HS train.
Interestingly, if a new HS line branched of HS1 on leaving the Channel Tunnel and headed NW, it could serve Shivering Sands, Stansted, and Luton before joining HS2 north of Aylesbury. That would give the airport direct high speed rail access to Birmingham, Leeds, Manchester, Brussels and Paris.