On contacting their support, I learned that they refused to use any other process. Also it became apparent that they had outsourced it to some other company and had no insight into the process and so no way to help. Apparently closing one's account will cause an escalation to a team who determines where to send the money, which would presumably put some human flexability back into the process.
(In the end I was able to get their web app to work by trying several other devices, one had a camera that for whatever reason satisfied their checks that my face was within the required oval etc.)