Statement of account does not show the structure of your tax, that is true, BUT:
There has been a change in the way NI contributions work and both Class 2 and Class 4 contributions of self-employed individuals are now being collected together with income tax, as opposed to the old system where we used to get 2 bills every year PLUS the rest of the contributions once you submit your self-assessment. And the Home Office teams know that. You could also point this out in the statement that supports your additional documents.
So if you have earned enough to hit the threshold and started paying NI, the HO caseworker will know that NI has been collected together with your income tax bill.
You can also submit a printout from your Self-Assessment oniine account. That will show the exact amount of NI contributions paid.
The point of the HRMC letter is simple - it is an official document issued by a Government body whereas the printout is just a printout, if you know what I mean.
If I were you, I would have made my best to play on the safe side. If a governmental organisation wants to see 1 proof, send them 2. A letter on HMRC headed paper, sent by an HMRC caseworker is indisputable evidence. HRMC's letters usually begin with 'Dear Mr/Ms xxx, We received a phone call from you on such and such date and you asked us to send you Statement of Account." HO can't dismiss such evidence.
Technically Nationality Directorate could cross-reference everything, yet they ask for all these things.