Of course: this is how synchronous AM detection works.
They generate a clean, in-phase carrier and use that to
demodulate the incoming signal.

Laura & Joel-

I was just thinking that. I recall there used to be (may still be) a shortwave
broadcast station just above the 15 Meter band, that broadcasted with a partly
suppressed carrier. Apparently the SWL community has access to equipment with
Synchronous detection capability.

Listening to it on an AM receiver, it could be understood but was distorted.
It was easy to tune it on a Ham transceiver as if it was SSB, but tuning was
critical for music to sound good.

Addressing the original question about direct conversion, what if you used a
temperature compensated oscillator with a phase-locked-loop? If the assumption
could be made that the broadcast station will not drift, then you could tune
each station dead on with an RIT-like adjustment of the oscillator.

73, Fred, K4DII