Voltage range shifter from 0V to 3V to -/+1.5V

Your test setup is basically a follower, which tends to reduce the effects of gain spreads. While this is good in production, it's not what's wanted here. In your job, the idle current is known, so you can set up a common emitter stage to pull the idle current. Also, the \$V_{CE}\$ the discrete op amp provides to the transistors can be determined. So you are setting up your test transistor under the same DC conditions that it will see in the discrete op amp. Now use a standard common emitter circuit with NO negative feedback. This implies using a fully bypassed emitter resistor. Now you will see more variability when you feed in your AC from your signal generator. Picking a pair will now be clear.

Low signal to noise ratio may cause glitches if the device can support sufficient frequency transitions. Also additive noise causes phase noise on zero crossing limiters.

Hysteresis is generally added when the noise cannot be controlled using positive feedback R ratio. This doesn't eliminate the phase noise but will move the crossing threshold in the opposite away from "zero" each time the output hits the peak. Often 1% is considered reasonable for some analog signals 10% for others and 33% for noisy logic interfaces.

For single supply amps, the "zero" crossing is set to Vcc/2 by some method.

One can even use CMOS inverter logic as a limiter (aka slicer, aka zero crossing detector) if the signal is AC coupled and use a self-biasing high R value (1M) as negative feedback.

The other thing about Op Amps is since they have so much gain and high order delay effects they must put in a cap inside to make the Op Amp essentially a 1st order filter for open loop to make it stable with unity gain.

Comparators on the other hand do not have this compensation cap so they work much faster as zero crossing detectors and ECL comparators work over 1GHz as well as current mode logic but using differential current with zero crossing current and differential load resistors.