Seems all too complicated. An SOP that gives the first officer full responsibility to initiate a rejected take off as PF if he considers it necessary, has serious implications legally and for flight safety. Is the captain merely an interested spectator while this happens? The legal eagles would love it.
It does seem to be SOP in a large number of airlines who have no legal problem with it... Why should they? With appropriate training and qualification, as in most human endeavour, there is little issue.
Some operators have a touching faith in the ability of their first officers to make instant right decisions and if that means they have the authority to order the captain to reject a take off for whatever reason, so be it. It is a recipe for disaster caused by confusion about who is running the ship at the time a decision to reject is made and then action initiated.
Maybe they have this faith because of the training specified above? If you're talking about a third world man-and-boy outfit, then it might be a valid point. In a normal airline, SOPs give areas of responsibility and interactive briefings cover the rest. There should be NO confusion over what happens in an RTO situation as this should have been discussed and agreed before takeoff - if this wasn't done then it's the fault of the PIC for allowing this situation to develop.
I don't think there is a "right" or "wrong" way of handling RTOs in terms of PIC/FO duties - if it works for your airline, your crews and your regulator, then fine.
What might be a long-term problem is (as a company) treating your first officers as lower forms of life who are incapable of making important decisions or handling the aircraft properly. When they eventually become a captain, where will all the reasoning and skills miraculously come from? One day not worthy, the next standing in judgement...