In the very least, the radiocarbon dating of the re-weave repair work sets the very LATEST possible date for the shroud itself; in other words, the shroud would already have to exist, and be old, at the date of the fringe re-weave. If the re-weave was performed in the 1290s, that means that the shroud is significantly older. It doesn't tell us how old it is, but it sets the "latest possible date." And that, in and of itself, is significant.
Is it the burial shroud? I don't think we can say. IF they ever let a test be performed on a portion of the shroud that wasn't subjected to re-weave work, and if that material dates to the 1st century AD, we still couldn't say -- for certain fact -- that it is the burial shroud of Jesus. However, it would make such an identification not only possible but, indeed, rather more likely than not; It would pre-date the relic-mania period of Christianity by almost 300 years.
I think the whole concept of religious relics is creepy. At the Cathedral of St. Anthony of Padua in Italy, for example, his tongue and larynx are on display. Lovely.
I don't think anything will prove conclusively that it's the shroud of Jesus until Earth and sky stand presently at God's great judgement seat.... and Jesus yells, "Gimme back my bathrobe!"