James T Kirk wrote:
The only even remotely possible reason he could have for destroying the Enterprise would be to stop word from getting out of what he had done, but would a Starfleet Admiral really behave this way?
There's your answer right there.
The admiral was not just trying to erase a really bad mistake. He was a warmongering sociopath that was itching to have an excuse to wipe out the Klingons.
Star Trek has a history of insane or evil (or both) admirals. Even just counting the "classic" movies, you have Cartwright and Dougherty, one from each era.
As TVTropes so succinctly summed up their list of insane admirals
Really, the list of admirals in Star Trek who are not either insane or talking heads giving generic orders off a viewscreen is awfully short.