I see the part about the BJT having to charge to a lower voltage. I also get now that although the BJT C-E diode voltage drop is higher than a MOSFET's D-S resistance voltage drop for a given current, the BJT is still a diode while the MOSFET is still a resistance, so the BJT can charge up a load capacitance to a higher level much faster since it's like charging through a diode (but the maximum charging voltage will be less than what a MOSFET can get). The MOSFET on the other hand is like charging through a resistor so that the charging voltage can achieve full supply voltage, except it levels out pretty quickly and takes longer to get there.
I was always aware of quasi-saturation, but we never looked into it anymore in class. I didn't know it only applied to BJTs (and IGBTs) and not FETs.