I think Ted has his team of evaluators and brings McCarthy and the coaches into the mix to define the kind of players they need at various positions.
Ted and his staff relies on the homework they do to come up with their own version of a board and it's Ted's decision when it's time to make the pick.
So I think Ted puts the 80 best players he can get on the field in the spring and from there McCarthy and his crew determine the depth chart based on the performance of the players.
My guess is that McCarthy has the most say when it comes down to the cuts until the roster is finalized. After all McCarthy and his staff knows better at that point when the players are on the field who he needs to keep.
I really doubt when MM hands the list to Ted that Ted says "no, we need to cut this guy instead and keep that guy". I think by this point it is going to rest mostly with McCarthy.
"The train is leaving the station."