Yes, that was often given as the reason, but think about it.
It was fundamentally wrong to design HS2 as a scheme from London to Birmingham, with add-ons to Manchester and Leeds. The correct design should always have been London to Manchester/Leeds as all or nothing, because that is what the UK has always needed.
The route never needed to serve Birmingham as top priority, because serving Manchester (and thus Liverpool and Scotland) removes many trains from the London to Birmingham route and thus inherently improves capacity here.
Your "correct design" is what was designed. A single, integrated high speed Y-shaped network that goes pretty directly towards the North West with Birmingham on a spur.
However, it was always going to be needed to be phased. For instance, there is not enough supply chain capacity to build the whole network simultaneously.
Phasing means prioritisation, and once the London end is prioritised, it's always going to make sense to deliver London-Birmingham first, because to build past Birmingham to reach Manchester frankly looks mad and it would be politically "very courageous" to try and build a railway through the West Midlands that did not serve its biggest city, especially since one of the key reasons why Birmingham's economy is believed to have performed quite poorly is to do with poor transport links.
HS2
does need to make serving Birmingham a top priority. Yes, HS2 will release capacity south of Rugby, but the railways around Birmingham, as well as New Street station are full, so building your proposal doesn't
inherently improve capacity. Capacity is constrained by the 2-track Coventry corridor etc.
Upgrading the existing network (e.g. 4-tracking Coventry to Birmingham, completely rebuilding New Street) would mean years and years of disruption and weekend closures, and would also cost billions, and you'd then need to do something similar the other side to address Birmingham-Manchester/North West Capacity. This forum is littered with threads and posts proposing or decrying ideas to improve capacity/frequency/journey times. They are not addressing an imaginary issue.
It does not help that West Mids to North West connectivity is also poor and extremely reliant on the M6.
These are a big part of the reason why, despite cancelling HS2 Ph2a, the Government also cannot escape the logic that it needs to build something like it.