The tech support people were right in requesting that you use the system from the internet source - namely the phone point. This rules out any problems with the wiring and/or modem in between the phone point and your PC.
As for "firing off a letter"..... I dont see what TalkTalk has done to deserve a slating on this forum. From what you have said, they seem to have done what they can.
As an MCDST, if I took a call from someone who "couldn't be bothered" to arrange using the phone point for the purposes of a test, then I would have had no problems in letting you leave the company.
This country has gone mad. If more people checked their OWN equipment and wiring, then half the posts in this forum would not be necessary.