In simple terms, based on experience. If your voice line is noisy when your broadband modem is connected, then the problem is the responsibility of the ISP. Apart from the fact that it makes perfect sense, Openreach engineers do not have the ability to carry out broadband diagnostics ( DSL status, Woosh tests etc) they have to call BT wholesale to get these done. So if you have an engineer out on behalf of your voice line provider, they are most likely not going to be of much help, especially with a somewhat illusive fault such as this.
I think that we have gone past the point where your own equipment could be considered as being at fault. From what I have read, I cannot be sure how much involvement your ISP ave had so far, but one thing I would strongly advise you to do is request a detailed list of all the engineering work which has been carried out so far on your line. Then ask for a no BS assessment from your ISP as to what work has NOT been carried out so far. They will have seen alot of faults in their time, and will most likely know all of the procedures which BT use. If you are not happy with the answer, then take your case higher within the ISP.
With faults such as these, there is often quite alot of buck passing, which is instigated purely by the nature of the fault, which is sitting rather inconveniently on the boundary of responsibility of the ISP and voice line provider.
And I agree with the last post on this thread, why not let us know who the ISP is? I personally am not one for bashing companies in public, but know for a fact that alot of ISP employees frequent these forums, it might lead to a helping hand, but of course, it is your choice