And, detecting it without a BFO*, results in distorted audio: the envelope is phase-sensitive, so you end up with something like abs(x(t)) which is a nonlinear function of the signal x(t). Voice will still be intelligible (if poorly?) but overall fidelity, obviously, isn't great.
*Beat frequency oscillator, basically reinjecting the carrier, and by way of the detector, mixing it with the sidebands. So, if it's not perfectly centered, you get split tones, e.g. a 1kHz signal shows up as Fc +/- 1kHz; if the BFO is off by say 100Hz from where Fc would fall in the IF chain, you get a new carrier Fbfo with sidebands of -900 and +1100Hz, detected as tones of 900 and 1100Hz. And, I suppose, tones of 200, 2000 and etc. Hz because the sidebands mix with each other too.
Which also means you can still use a PLL to lock onto the original (suppressed) carrier. So it's not too bad, but it is more work to do that of course.
SSB of course intentionally omits the carrier, and also a whole-ass sideband, so no double-mixing occurs, and so you don't get pilot tones or beat frequencies, just the warped Mickey Mouse voice when it's not tuned quite on center.
(Heh, I think you know much of this already, so, just putting this out there for anyone who doesn't know, and is curious what the full picture is.)
Tim