So, I have been chatting to T-Bank customer service today to get to the bottom of this matter. I am travelling to Russia in a few days' time so the issue is of immediate practical relevance to me.
It is certainly possible in principle to both register and verify a Gosuslugi account via T-Bank, with no need to visit an MFC office. The T-Bank app crows about this possibility! It says "You can register on Gosuslugi through us" and "We'll verify your account immediately, online and without visiting MFC."
This option only appears in the T-Bank app, not the personal cabinet on the website - if you try there it says it is "impossible for a non-Russian citizen", but it is in fact possible if you use the app as the (human) customer service assistant explained. As the app is not available in the Google Play store, she sent me a link to download the app onto my phone.
The first problem, however, is that if you press the "Sign up" button on the app to register for Gosuslugi, an error message pops up saying "Error. You can only sign up on Gosuslugi with Russian mobile number. To change number please contact the bank." So back to the classic Catch-22 (no Russian phone number without Gosuslugi account, no Gosuslugi account without Russian phone number!)
That is not the end of the story, however. I went back to the customer service chat and a human assistant sent me a link that I could use to complete the registration process without a Russian phone number. This involved typing in an email address where Gosuslugi would send me a link to complete the process. The T-Bank assistant was on hand throughout to walk me through this process.
But - alas! - this was not the end of the technical problems. The verification email I was supposed to receive from Gosuslugi never arrived. I believe the reason for this is that my email is an Outlook account and Microsoft appears to be blocking messages from the Russian government. This is quite a pain and useful to know. This would presumably also have been an issue if I had tried to use my Outlook email for verification at MFC.
To get round THAT problem, I set up a brand new mail.ru account. This time, the email from Gosuslugi DID arrive (so indeed this was a problem with Outlook accepting the emails, not a problem with Gosuslugi sending them). But I was still unable to complete the registration process.
The reason for that is that clicking on the link in the Gosuslugi email is supposed to open a Gosuslugi web page. The problem with that is that all Gosuslugi websites, like most Russian government websites, are inaccessible in my country (the UK). So I am going to have to repeat the process, beginning with the initial registration of a Gosuslugi account, when I arrive in Russia in a few days' time.
I will also try to complete the verification process at a Russian bank, as it is clear that banks have been accredited by the government to do this. But if further obstacles arise (most probably technical ones similar to the ones I have experienced today), then it is no big deal if I need to go to an MFC office, as there is one of those nearby.