direct conversion SSB (vs DSB-SC)
The whole point of a superhet is that you can do the IF filtering at a FIXED frequency.... A 3KHz wide filter tracking across say the 40M band with sufficient accuracy to pick off the lower sideband while rejecting the upper one 600Hz away is a BIG ask, less so at a fixed 455KHz or so. 

SSB really does not care over much about carrier phase (after all the carrier will ultimately end up at DC if the tuning is exact!), but you do at some point need to remove the unwanted sideband and that is usually most easily done (in a classical superhet) by mixing to a frequency that puts the passband of a crystal filter in the right place to pass either the upper or lower sideband. 

Modern SDR practise is often to use direct conversion to a quadrature pair at zero IF and then uses either weaver or a hilbert transform to extract the sideband of interest. The thing SDR has going for it here is that working in the digital domain we have both REALLY GOOD mixers (Multiplier blocks in FPGAs are rather linear) and exact gain matching when playing with complex numbers, both of these things are problematic in the analogue world. With analytic signals there is nothing magic about zero frequency, it is trivial to move it around as required.