Airbubba
Some airlines have the FO do the reject if he or she was the pilot flying and then hand the plane over to the PIC when stopped as seems the case with BA. Others have the captain always take control immediately when the reject is announced.
At BA can any of the flight deck crew members call a rejected takeoff?
At BA the captain can call "stop" for any reason if he/she sees fit, the co-pilot has a list of things he/she can call stop for, mainly but not exclusively the really critical stuff such as engine failures or any fires. Any extra pilots on the flight deck are usually briefed to call out anything abnormal they think the operating pilots have missed, but according to the Ops Manual are never supposed to use the "stop" word........
Whoever is handling stops the aircraft, handover to the captain (if needed) is done after the parking brake is applied.
As for recall/memory items in the QRH ...yes you're right, much reduced and rationalised over the years (and yes, in our case the move has indeed been to junk legacy procedures and align with those published by Boeing/Airbus. ). FWIW the evac checklist is now meant to be a "read and do" if circumstances permit.