Yes, HS2 links the cities you mention - that much is clear even to the most agnostic of us. The question is not what it does, but what the purpose of all those thousands of additional journeys will be.
The purpose of HS2 is to provide for those journeys, not to analyse their own purpose, other than to recognise that they are desired. It's called predict and provide and it's what the Government did with road building, until that became unsustainable. It would be quite possible to choke off demand by not building HS2 and at the same time to make the present routes undesirable by high prices, poor service and so on.
When you write of journey purpose, it's classed as either firm's business, journey to work (that's the Americanism 'commuting'), or leisure (shopping, tourism, visiting relatives, etc).
The CBR for HS2 and all transport schemes is based on what the predictions are for these uses. The journey time savings for each class are priced accordingly to their respective values. Travelling on firm's business is far more valuable than visiting one's aged aunt or watching a football match.
I presume that the HS2 reports (I have not read them all) contain the details of all these predictions and costings, and they must be held to be sound, otherwise opponents would have succeeded in their high court action, as one of their objections. They didn't, at least not on these grounds, so I assume that's the end of that. Going back to Philip Elliott's postings, I have long mislaid from memory what his original point was, but I understood he thought HS2 should have gone via places like Coventry, MK and Stoke (or was it that it shouldn't be built anyway?). Well, it doesn't go to those, neither does it go to Bradford, or Harrogate, or Huddersfield. Think of it this way; the M1 does not go to those places either, - so what does it matter, HS2 is just another transport scheme, just as GW electrification is.
Joining railways to HS2, just like joining roads to M1, can always be separately justified in future schemes, if the BCR looks good.
I simply do not understand what the debate here is supposed to be about, other than if one rejects the 21st century and all its opined unsustainable doings.