Will it? Crossrail to Bond St then tube will be about the same as HS2 to Euston then tube. Crossrail to Moorgate then tube will be faster than HS2 to Euston then tube.
Err, sorry, what? The station that where Euston gains a further 2 minute advantage over OOC would be the one that's slower via Euston whereas the other is about the same?
If you had said them the other way around then maybe I could treat you as a rational actor, able to be persuaded. As it is, I'm just showing any casual anti-HS2 people listening in that you seem incapable of simple logic, reading comprehension and arithmetic. And thus, when you say (completely untrue, btw) stuff like "phase 1 will only save 10-15 minutes", it should be clear that you have no credibility to pronounce that - even if HS2 were themselves saying it - as you can't be trusted to read data correctly, deal with numbers, or make sense.
Let's look at the time differences:
Bond Street/Oxford Circus: 8min vs 2min
Tottenham Court Road: 10min vs 4min
Farringdon: 13min vs 6mins
Moorgate/Liverpool Street: 16min vs 8mins
So the tube is 6 minutes (8-2) shorter from Euston than OOC to Bond Street/Oxford Circus, but 8 minutes (16-8) shorter from Euston than OOC to Moorgate (or Liverpool Street Elizabeth Line station*). The one which the last leg is 8 minutes faster via Euston is where it's more likely to be faster via Euston than changing at OOC than the one where the last leg is just 6 minutes faster.
With 5 minutes extra on HS2 to get to Euston and 2 minutes more interchange time there (though maybe there won't be) than at OOC, and it very much is, as Ianno said, "swings and roundabouts". When I said "a little bit shorter", I mean like an unnoticeable minute shorter. Which is what we get (with Euston sometimes quicker, OOC sometimes quicker). And given this is for the central London locations with direct trains to OOC, it's clear that OOC isn't going to be inherently better for central London.
*We'll ignore that you suggest getting on the tube after taking the Elizabeth line from OOC to get to Moorgate, when Liverpool Street Elizabeth line station will share a ticket hall with Moorgate station...