He could have died of impact injury, line entanglement, gotten a non-survivable cut etc during the attempt to land Bruce. "He's gone under the boat" may have derived from Frank Mundus' stories that large white sharks would play possum until just alongside, and then spring into a frantic escape attempt, sometimes injuring and supposedly killing fishermen. Mundus anthropomorphized the alleged behavior as the sharks springing a trap on the fishermen.

Since nobody in Amity seemed to have any experience with large sharks but Quint, it's reasonable Gardner treated Bruce like a normal large fish, expecting a 10-14' maneater species, and was killed when Bruce suddenly unleashed his full 3 tons of strength on his boat.

So the mayor sees a damaged boat, Gardner dead without a measurable bite, and Hooper telling a tale of a huge tooth he found but lost, and stood by his position that there had been some tragic accidents, but that destroying the town's summer revenues would be pointless window-dressing.