Electronic – How to determine BJT switching time

Saturation simply means that an increase in base current results in no (or very little) increase in collector current.

Saturation occurs when both the B-E and C-B junctions are forward biased, it's the low-resistance "On" state of the device. The properties of the transistor in all modes, including saturation, can be predicted from the Ebers-Moll model.

First, these are NOT voltage followers. They are inverters. There is no reason their gain magnitude couldn't be more than 1. In fact, I expect all these circuits already amplify, but since you're putting a digital signal in you don't see the amplification when the input is near 1/2 the supply. That's probably a good thing because leaving the input there would fry the transistors rather quickly.

While the gain magnitude is most certainly above 1, it is not predictable or controlled. There are various ways to get that, like adding emitter resistors with some feedback per stage.

If you want positive gain like a Schmitt trigger, then put two of your inverters together with a little positive feedback around the pair.