I should like to see the schematic of that modulator. It looks about as simple as you can get, and as such will have a degree of FM imparted to the signal as well as the intended AM.
I also have a pantry transmitter project in mind for when I get "a round tuit".
I have been very impressed with the AM sound from my 405 line TV modulator which operates around 50MHz. The oscillator is free-running, as is yours, but drift, even at this low VHF frequency is not a problem, and I can listen to the carrier on a communication receiver with the BFO switched on and the note is pure with near 100% modulation applied.
I see no reason why it wouldn't work at 1MHz with the appropriate tank circuit and a PA stage to put out a few miliwatts.
Uses an MC1374, the lower half of this cct
http://www.earlytelevision.org/405_modulator.html