I think it is well known that some ISPs take fault investigation far more seriously than others.
One approach is to think as if you're part of the faults team and provide actual evidence. On one faulty line we eventually discovered using the 17070 ring-back test that the modem lost sync every time that an incoming call was answered, but not whilst it was ringing. I have had routerstats running for about 6 weeks via logmein on my laptop on a line that has been problematical for months. After every "no fault found" visit I only have to monitor the modem whilst ringing the number with an answerphone to demonstrate the fault was still there. It has taken BT many elapsed weeks to cure the fault with a "Lift & Shift" today. Another very useful tool (I've used on another poor line) is to monitor the diagnostics page on a 2Wire HGV2700 modem which provides many useful measurements. In this case I take screen pictures in my house of the data which I can then e-mail to the ISP for further assistance.
As I am involved in multiple problems I have eventually been noticed by the local BT O staff as someone who can actually help them. Their tasks in this very poor performance area are not to be envied !
However it is perhaps unsurprising, particularly on difficult and intermittent faults, that BT O's systems are designed to restrict involvement with the end users, unless and until you can demonstrate that you understand the problem details and that often requires solid evidence.
Kind regards,
Walter