It doesn't even have to get to the point where phase comes into play.
If you don't have it "zero beat" the two sidebands will not only sound odd
(because they aren't converting down to their original baseband) but the
two sidebands will clash with each other. If the carrier is in the right
place, both sidebands will convert to the exact same audio frequency. If
the carrier isn't right in the middle of those sidebands, the two
sidebands will convert to different audio frequencies, so it will sound
awful, at the very least.
Single sideband can be mistuned somewhat, all it sounds like is osmeone
talking in a higher or lower tone than "normal". But if an DSB signal is
demodulated with the carrier not right in the centre, the two sidebands
converted to audio interfere with each other. If the carrier is right in
the middle, they combine properly.
Synchronous detectors were originally created because of this, to properly
demodulate DSB with no carrier. Which is kind of odd, since in more
recent times synchronous detectors have become relatively common in
shortwave receivers, yet some or many of those designs likely don't work
with DSB no carrier. It depends on where the circuit is getting the
information to control the BFO.
There was a time when DSBsc demodulators were shown in the ham magazines,
the simple ones would square up and then double the incoming signal and
then divide it back down to the IF signal. Which always resulted in the
carrier being right in the middle between the sidebands.
But a synchronous detector was too much trouble, might as well use an SSB
receiver and convert the DSBsc signal to SSB inside the receiver. By the
time synchronous detectors were known, the move to receivers for SSB had
already started.
Of course, DSBsc has other advantages beyond the simple transmitter, a
proper demodulator makes use of the redundancy the two sidebands to
improve reception. If nothing else, it gets information from both
sidebands, yet allows one to switch between the two if interference is on
one sideband. A proper DSBsc system is more complicated at the receiver,
but has some advantages over SSB (though narrower bandwidth isn't one of
them).
Various shortwave broadcast stations have played with SSB for
transmitting, and often have had to fall back to SSB with reduced carrier,
since if they are playing music, they need the carrier to tell people
where the local BFO should be set; otherwise the music converts to the
wrong audio frequencies, and sounds way worse than a mistuned voice SSB
signal. Or they could put the other sideband back, but eliminate the
carrier. That way the unneeded carrier isn't transmitted (so more power
for the sidebands) but the extra sideband provides the information needed
to insert the local BFO right in the middle of the sidebands for proper
demdulation.
Michael