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At the time of the testing (the 1980s) a considerably large piece was needed to test, which in the testing is destroyed. The curators of the shroud were reluctant to allow any of the cloth be snipped out, but finally allowed one place, off to the side of the image to be tested. Subsequent imaging and other non-invasive testing has shown where they tested was a very carefully woven in patch...evidently from the medieval times.

Serious testing would demand several pieces from all over the cloth—which at this time I think would only need a thread or so....yielding no holes in the cloth.

The Vatican should allow it to be retested except, there’s really only a practical downside. If a better uniform test proves its medieval, well, then it’s value as a relic becomes useless. If it’s shown to be 1st Century, the skeptics won’t care, and still won’t believe. As it is now, believers believe, and skeptics don’t, and who wants to rock the boat?

It's possible, as you said, that the Vatican is awaiting a better test that would prevent any further destruction of the Shroud. That would be my assumption.

If it is indeed the Shroud of our Lord, waiting a few years or so seems judicious.

The Vatican should allow it to be retested except, there’s really only a practical downside. If a better uniform test proves its medieval, well, then it’s value as a relic becomes useless.

One little problem: The Shroud survived two separate fires, which could have affected the amount of carbon isotopes in the cloth. Thus any radiocarbon testing at all, no matter the result, is questionable at best. The issue is explained as part of a discussion here.