Emitter Followers do not saturate. There is a reason for it. Since the load current is pulled by the emitter it can never rise above the base voltage for in-order to get Vce<Vbe. It would require a Vb>Vcc to saturate Vce.
The popular old L298N is rated for low voltage and 2A but not together. If you used a single supply Vcc =5V with a 2A load the voltage across the load worst case is almost zero, meaning almost a short circuit is needed to get 2A. But for low currents a few volts across the load is possible.
It is an emitter follower high side and common emitter low side in an H bridge while both are used to drive a differential load. There is a large drop allowance on the external low side current sense resistors which must be minimized preferably 50mV max then amplified for feedback.
BJT emitter followers have higher linear operation and better hFE without the saturation effects but also with a higher Vce drop than Vce(sat).
BJT saturation depends on the CB junction no longer being reverse polarized and the Ic current rise and voltage drop of Vce less than Vbe.
This apparent forward conduction of collector-base also reduces the maximum linear hFE current gain into this non-linear mode, as the collector is no longer a high resistance current source but with lower current gain and becomes a low resistance voltage source or switch. Whereas emitter followers have the emitter drawing a base current Ib=Ic/hFE ( and Ic~Ie)
You won’t see these with complementary shared collector outputs because the risk of shoot-thru across Vcc+/- or gnd will short the supply with inductive loads unlike CMOS which share common drains. CMOS or complementary power FETs are easier to control the required “dead-band” or dead-time where both drivers are turned off or partially conducting yet must be protected from flyback voltage.
Generally, Vce=Vce(sat) is often defined by Ic/Ib=10 at some nominal rated currents. When you see transistors rated for Ic/Ib = 20 or 50 it is only because their linear maximum hFE is > 10 greater than these ratios. There exists hFE’s > 1000 rated at 50:1= Ic/Ib but this comes at the expense of other parameters like Rce= delta Vce/delta Ic in saturation and also BW reduces for possible GBW products so they are not common for switching transistors. Diodes In has many patents on on transistor design to improve this and rate their best devices with low Rce in mOhms vs the typical 1 Ohm for a PN2222A but this comes at a great cost,so you won’t likely find it in motor driver IC that need fast dV/dt.
I recall the audio transistor 2N5088 has these high hFE’s with gold doping but also limited to audio Bw applications.
Since Vce is created by the emitter load current pulling down the base current, and the base voltage always being slightly lower than Vcc, Vce can never go lower than Vbe or in other words the CB junction can never see a forward voltage and thus Vce never reach Vce(sat). Thus these “old school motor drivers are only useful where the Vcc is much higher than Vbe to avoid the dropout voltage that is also common the BJT Op Amps. With emitter follower Darlington drivers, it’s even worse with more voltage drop and motor drive power-loss with two BE diode drops.
Thus unless cost is an overwhelming bias to choice, FET bridges are preferred for Vcc <= 12V for lower heat rise and greater efficiency.
Of course you can always sink that heat and raise Vcc to generate a voltage to match the motor limits but this is a big cost-performance tradeoff for low voltage motors.