It was tested 1988. IIRC only once before. What is the problem in agreeing testing standards acceptable to all, with blind controls, as an ongoing research project? Sending off nine samples from one tiny area of the cloth, with no controls, is not satisfactory by any stretch of imagination. Bring in the archaeological scientists. Let the Vatican scientists have a bash.
Let's have transparency!
The 1988 protocol was agreed on by all participating, had blind controls (one of the labs actually ashed all samples to avoid anyone being able to tell the controls from the Shroud due to the visual appearance of the weave), and three labs independently had quite good matches in their datings. The appropriate scientific experts were involved at all stages, including Vatican scientists, and a great deal of effort was expended to be certain that the tested cloth was representative of the whole by all experts involved.
As to your last point: Think it through- was it the scientists who were and are unwilling to test even more of the cloth, or the religious order who owns it? Who do you think preferred to ideally test pieces from all over the cloth and who preferred to test an localized piece away from the image itself? If you have a (what you consider) sacred artifact, just how much of it do you think you are willing to chop up, deface, and burn for an isotopic dating? This is true of almost any precious object: "Good news Mr. Smith- you really did own an authentic diamond. Unfortunately we destroyed the entire stone to be certain that it was all truly diamond..."
Again, these factual points are all discussed in this very thread. It may seem lengthy, but it is very repetitive. I suspect that if you wish to make suitably informed comments, you can probably come up to speed on the thread in a day or two.