What is conspicuously absent from the dialogue is any attempt to confirm that their airspeeds were indeed "normal". They were not. Despite having multiple warnings that are explicitly stated in AD 2018-23-51 (the 737 AD published post Lion Air), this crew made no apparent attempt to determine this crucial fact. If they had done so, then performance of the Airspeed Unreliable non-normal would have been anything but discretionary.
AD 2018-23-51 does not address Airspeed Unreliable. It is concerned with identifying a Runaway Stabilizer condition. It merely mentions symptoms that may indicate this condition. It does not direct pilots to perform Airspeed Unreliable actions in the event of stick shaker.
Yes, there was a slight difference between ASIs during climb out but at that stage was in the order of a few knots and practically speaking would probably be unnoticed in the dynamic period after liftoff, particularly in the absence of any alerts.
you are retrospectively creating a process based on assumptions that involve hindsight.