Even the Starfleet emblem was a cloth patch. If you weren't careful putting on your maroon jacket from TWoK, you might take a frigging eye out with all those pins petruding everywhere ...
Fletcher actually thought about that for TMP, too. Kirk's two-tone uniform had a metal pin, but that was only supposed to be for ground/shore duty. On a ship only the sewn-on badges would be used for safety reasons.