This dates back to FAR 25 A/L 42 (if my memory serves me correctly) ... it has nothing to do with continued takeoff .. only the reject.
Prior to that amendment, there was effectively nil fat built into the certification rejection exercise .. and, surprise, we had the odd reject on critical runways end up off the end of the strip. After a lot of Industry pressure, the FAA moved to introduce a nominal delay at V1 to account for things like startle factor, scatter in pilot reaction times and so forth.
The usual manner of running the testing is something along the lines of
(a) fail the engine at the appropriate point just short of V1
(b) correct the yaw on rudder
(c) do nothing else until the delay period has passed
(d) execute the prescribed reject sequence
There will be a modest speed overrun but that is part and parcel of the measured reject distance.
The delay deal is intended to be transparent to the pilot .. ie the intention is that the pilot follows the flight manual liturgy and, in the reject sequence, does the tapdance as expeditiously as is practicable ... there is NO intention that the later certification standard provides fat for the pilot to dilly dally with initiating the reject.