I did explain that in an earlier draft, and that the OP and many readers may not know or if they did may have forgotten, but it began to read like a treatise

.
I cut it to the essential. My point was that the CS staff and the engineer, who may or may not have been from Openreach and in any case could be a recent recruit, could have been getting confused. For instance, following
"Occasionally the ADSL drops too, perhaps for a few minutes, perhaps for an hour, which isn't ideal", we have:-
Last time my line dropped for a significant period of time I reported it to A&A as a PSTN fault; an Openreach engineer came round but couldn't find anything wrong (though acknowledged the existence of music through the corded phone) so nothing changed.
That almost certainly means a phone engineer was sent, not a broadband engineer. When the problem was the broadband had dropped for a significant time.
Then we have:-
I've already done the testing through the test socket etc., and indeed reported it to A&A as a PSTN fault as I have my phone line through them too.
Uh uh!
We don't really know what the OP said to CS, but the odds (in my opinion) are that a phone/voice fault has been reported to Openreach. The wrong sort of engineer was sent. Would they be bothered about music on a line that didn't have a phone service, given that the engineer could well know nothing at all about the effect that would have on broadband?
Zarjaz's post was relevant, as it tends to annul my theory any such lines are terminated on a service that plays music, as at the moment Openreach and particularly BT Wholesale that I think the OP is on (I think their ADSLx service is only on BTW not TalkTalk but I could be wrong) do not generally do a line-only connection for broadband.
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