As a mode for communication, as you say, it's pretty redundant and inefficient. I've never heard of it being used seriously.
Occasionally people might hold a morse code practice class and transmit the sound of morse code on a 2 m FM channel. Presumably because FM radios are what the students have, and perhaps they're not yet allowed to transmit on HF. This idea could extend to AM... maybe.
But one place you will regularly hear morse code as an amplitude modulated tone, is on a Non-directional beacon. These are transmitters set up on long or medium wave, 190 - 535 kHz. An aircraft uses a direction finder to get a bearing to each beacon. This allows them to locate themselves and fly routes from beacon to beacon, pre-GPS and still today without GPS.
Each NDB in an area is assigned a different frequency, so the pilot can dial in which one they want to check, but they also identify themselves with a few morse code letters. These are amplitude modulated for identification, because they don't want to turn the carrier off at all. DF works best with a continuous carrier. The signals are strong, which is required for DF, so no problem with efficiency. And an AM tone is easy to demodulate and decode.
I used to scan around for these in my car (its radio went down to 150 kHz) and try to figure out which airport was which, based on the cryptic two letter codes.