Another terminal station is a massive failure. Another station detached from the transport hub at New Street is a second massive failure.
The Zurich project is well worth a read. It translates very well towards Manchester or Birmingham.
Crossrail 2 on the other hand, not committed, will need about £40bn to clear up the mess created by repeating the failure of centuries past in dumping 100,000 extra people a day at a terminal in Euston.
Let's assume that we did build HS2 with a through station in London where would it go to?
Even if it did do somewhere else could we even be sure that it wouldn't still lead to 100,000 passengers being discharged at Euston station. In fact it could well be that it would be higher than this if it means that you end up with a lot of people coming in from the opposite direction as well and/or travel between two London stations.