I have now tested this every which way and more, and it seems to be working.
The issue was entirely on their side, as you will have suspected.
They had themselves told me to set the setting they referred to in the way I did.
And then when I brought up this further issue, rather than looking more closely, they tried to blame me (us) I think on the assumption that I had recently switched to PHP 7.x
Both the senior support agents I dealt with then seemed to believe that all they had to do was the test emails you established and that I directed them to was to use some form of default value garnered simply by loading the page. They did not seem to realize that they actually had to get their hands dirty and enter some input values!! Amazing.
And as soon as they entered some input values and got the SMTP debug information, it of course became as clear as daylight to them what was happening, and I have adjusted the self-same setting they originally told me to adjust.
As above, all testing that I have done has achieved the result we want, and I believe this is now fixed.
However, I am telling the email server people that it is not to be regarded as fixed until such time as I have done a mailing to my list. Just so they keep the ticket open for a couple of weeks.
The only issue that MIGHT come up is when I do such a real mailing to my mailing list. I really do not think that that is going to affect anything. I therefore think we can regard this matter as concluded.
If there is indeed some issue when I do a full mailing, I will start a new ticket with you simply because it will be easier, and we will both know what to focus on.
Thank you very much for your kindness and help, and I am truly sorry we went round in so many circles when the issue all the time was one that should have been resolved in a far more timely fashion by the mailing service.