Does it actually need *rebuilding* entirely? What's wrong with the staged HS conversion proposal which wouldn't leave it down any more than 3 platforms at a time?
It would be harder to operate while down 3 platforms, but I reckon it could be done if servicing was moved off-site - VTs do tend to lay over for rather a long time.
Think of London Bridge, just several times more difficult - the more trains and passengers you can move away from the construction site, the quicker and less costly the new station will be. I've been involved with building motorways and major roads over the years, the more you can build away from the daily traffic the easier life becomes - ease of construction lowers construction costs.
With regard to passengers being displaced, you wouldn't just move random trains, the methodology may include redirecting services from (for example) Liverpool & Wales to a different site, then the Manchester services to another - it's not going to be an easy task to redesign the timetabling but let's face it, the disruption will be for nearly a decade and the routing will change after completion anyways.
Re the laying over - give a yard and the other person will take a mile - whilst I accept that some lay over time is necessary, platform space will be at a premium and reviewing operational methods are a way of reducing that lay over time, could be using extra drivers to move stock out to service areas away from the station, or extra staff to get the cleaning and routine checks done quicker.
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Again, it would be a nuisance.
Nuisance or not, some pain will come from construction of HS2 - you can't just do the works overnight for a decade. The key is to make the changes compatible with the end product - There is a clear concern from TfL and others that banging 18 train loads of HS2 stock into Euston is going to need substantial works to the tube stations serving Euston, cause upheaval in bus and roads based links.
Whilst traditionally central London was the target destination for the mainline services, times have changed, passengers arriving at Euston are heading to a variety of different places not just in central London, but across the SE of England and beyond - simply accepting Euston as "the" place to terminate services from the north is backward looking and doesn't acknowledge the economic spread of businesses and tourism out of the city centre, nor the casual traveller who has an ongoing trip from London - Thameslink, Crossrail 1, 2 and hopefully 3 will go someway to resolving this, but further work is needed going forward.