It is likely that any future high speed lines would be constructed mostly underground anyway to avoid the endless campaigning and problems with locals along the route.
It's more expensive in terms of construction costs, but the route might actually get built and you won't have to spend many many billions on land purchases and the like......
But also why are you not doing anything about the South Coast?
It's demand is more likely to be able to saturate a new line than any of the others you propose here
Given that the likes of Crossrail 2 and Southern Approach to Heathrow aren't certain to be built and they would enable several new services to be operated in the very area in question, then I don't hold out a lot of hope for any of the proposal to happen, much as I think it would be a good idea.
Crossrail 2 should have been started half way through the construction of the first Crossrail, as the extra 8tph would be well used upon opening, even in the off peak periods. Maybe not full (although some could be) but certainly busy enough to justify being run.
As such there's almost certainly enough demand for a new line/extra tracks to the likes of Brighton or Portsmouth or Southampton. Whilst if you built a high speed line serving all three it would certainly be well used.
A new High Speed line running London, Brighton, Portsmouth, Southampton and then back to London by a second route would mean that no one place was at the end of the line.
It could also allow services to run fast or of London to a junction to then serve other stations/lines. For instance running fast to near to Basingstoke and then calling there and then at places along the WofE line without slowing down services on the main HS line.