Chekhov leaves a phaser behind

"Likeliest outcome is that the guy who caught it just threw it away and it got buried in a landfill, or it got put into an evidence box and was thrown away years later."

That's not a likely outcome at all. It's an obviously high-tech device of unknown function (presumed to be a weapon of some kind) that was being carried by a Russian who had gained unauthorized access by unknown means to a US Navy nuclear-powered aircraft carrier during the Cold War. They obviously would have assumed he was a Soviet spy and his unknown device would have been an object of great interest to the powers that be.

"Because no case ever would have been filed or prosecuted, as there was no serious crime and the defendant disappeared from the hospital, and as far as anyone on the Good Ship Enterprise knew, Chekov was just some random nutball who wandered in off the street."

The civilian criminal aspect of it would have been irrelevant, and he wouldn't have been viewed as "some random nutball who wandered in off the street." He was Russian and carrying an unknown, high-tech device that he tried to use as a weapon, and you can't just wander in off the street onto a military ship, so how he got past security measures, completely undetected, and what he was doing to cause a power drain, would have been of great concern.

This would have been a matter of national security and taken very seriously, especially once they took that phaser apart and saw technology that's well beyond any known technology of the time. It most likely would have ramped up the Cold War due to the apparent evidence that the Soviets had leapfrogged ahead of the US in weapons technology. It would also be assumed that if the Soviets had handheld versions of this advanced, unknown technology, they would also have much more powerful scaled-up versions as well.