BJT push-pull modification
Mainly an aside: Remove D3 and D4 and see what happens.
These are output stage antisaturation diodes that stop the output transistors being driven into saturation so that they recover more rapidly on turn off.



Unless I am missing some inobvious linkage It consists of 3 separate circuits which are attempting to model the same thing in slightly different ways.

These are not follower circuits (according to my brain). 



This may not suit what you want, but try:

Remove C3.
Disconnect R3 and R4 from drive line and connect them to out_sat.
Monitor out_antisat
Trial.

Signal is inverted from what you had.



Next:

Add resistors from out_anti-sat to Q1 and Q2 bases.
Size tbd.
Say 100k to start.
These provide positive feedback.
Trial.

Report.

If you want rapid transition across the centreline then making each half a true Schmitt trigger (2 transistors per half) is easy and liable to be effective.