The remains don’t have to be destroyed; they just have to be unidentifyable, or not found.
Someone may be shot by a gun in a far corner of the battlefield, his body not found until decomposition made it hard to identify him. Soldiers did do things to identify themselves before dog tags became standard, but they could get lost. In the Civil War, you might write your name on your clothes, only if someone stole your coat before your body was found it would do not good.
Similarly, if were aboard a ship that sunk in deep water, your body wouldn’t be found. If all the people who could identify you were unable to do so (like if your unit suffered massive casualties), no one would be able to identify your body.
Nowadays, with DNA tracing, it’s possible to identify a body even if the person had been dead for years. But in older wars, you needed to figure it out before decomposition made it tricky. If anything, I’d assume there are fewer MIAs now than there were in past wars.