You actually got them to admit they had a problem?
Last time I had to look into a TT issue, they blamed the inability of the customers router to log onto their network on the router... The fact that it had been working fine, and nobody had changed anything for years wasn't the point. The router was customer supplied, so that must be it.
Even explaining I'd tried two routers, both of which has negotiated a good speed with the exchange, but had failed the log on (which occurs afterwards), the problem couldn't possibly be their end... Oh no, it'll be both the routers I used, or maybe BT will have to come out to check the line...
Yes, they were still blaming the physical line, or router, or anyone by themselves.
Took 3 days of calling before we got to someone who realised they weren't talking to a complete idiot. Even then it took over 24 hours for it to magically start working, and they never placed the follow up call to let me know either.
Nobody should ever use a TT email account, you're not going to want to stay with them for a second longer than necessary.