Because you don't have noise on the phone at the test socket, TalkTalk should be treating your fault as a broadband fault and not as an ordinary phone fault. Basically, you have intermodulation between the broadband and voice frequencies resulting in increased error rates and SNR margin drops, all due to an HR fault no doubt, which as an interoperability fault can only be fixed by a broadband engineer. Either TalkTalk doesn't understand this or they are using "no noise on the line at the test socket" as an excuse not to do anything.
As for this business of testing at the test socket with engineer installed FTTC, I was never asked to do this by BT Retail during my own fault solving as I believe they consider the filter to be part of the NTE5 and Openreach property. Of course, I did it anyway.
It sounds like the fault can be mitigated somewhat by a lengthy phone call (increased current heating faulty joint?), therefore try this: turn off the modem and make several long phone calls; turn the modem back on. Your connection may now work a bit better (but for how long?).