I apologise that I so often fail to be clear.
What I thought might assist was if we created a User Record explicitly for Fastmail.
They can then explicitly try to send and generate their own emails using the "send me validation link" option.
(I.e. by using that test email address they gave me and so that they could log in for themselves as an authorized User pre-existing in the database in that specified user Record created for them would allow them to see the issue directly and for themselves).
I had thought that an attempt to generate a validation email would be sufficient evidence to them that nothing was sending anywhere, with the debut record giving them information for themselves.
My fear is that they are going to waste YOUR time -- mine is far less valuable -- sending repetitive instructions through me to you about obvious measures you have already taken, thus recreating the wheel for days on end while we get nowhere.
If, on the other hand, they pressed that "send me a validation email" for themselves, I just thought they would see the problem straight away.
Please find my credentials for the service in the private area.
I hope that works, and that you indeed see something to resolve this situation.
Thank you for the offer. I hope it saves us both some time.