As to SSB AM, (called "AM Equivalent" by the military) some radios will do this, but others won't.

The military surplus GRC-106 for instance does this, and the final is a linear amplifier with associated efficiency issues. To the listener, it sounds like AM, but they can tell that a sideband is missing if they tune around.

The Johnson Viking for instance does not do it, and the final is a class-C stage modulated by a class AB2 push-pull stage.

So some gear just can't do it. I don't prefer linear amplifiers as final stages for AM, but it's an opinion.

There is nothing wrong with either mode. SSB AM can be used in the AM nets along with DSB AM.