Well, EA so far has been less than helpless, BUT, I believe I'm getting closer to the answer.
After spending almost a year on and off trying to run this issue down, I had a strange thought that maybe the issue wasn't with my network or hardware or installation, but my account. I had my brother sign into his EA account on my PC, same hardware, installation, network, and wouldn't you know it? It instantly connected and works just fine.
So, the only thing I can think, is that EA has effectively banned my account from playing BF1. I am not sure why this would be, as I have never cheated, and furthermore, the EA "Penalty History" for my account (https://help.ea.com/en/help-tools/penalty-history/) is completely blank, so I guess they've effectively shadow banned me from BF1.
This doesn't really help me much in terms of being able to play the game again, as now it feels entirely in their court, but I have once again opened a support case, now armed with this knowledge, so hopefully I get a response this time, ideally with some answers and maybe, if I'm very lucky, a resolution.
I'll update this thread regardless of what comes next, for posterity. I will be royally pissed if EA isn't interested in helping, and I lose my 454 hour account and am forced to create a new EA account and buy the game for the third time (yeah, I also bought it again on Steam on sale at some point, in the futile hope that might fix it,) just to play it again.
Wish me luck!