The problem with that approach is that I don't think you would get first, second, third or fourth choice in some cases, especially when you are looking for players to improve the squad. I don't think placing deadlines works for clubs who don't really want to sell their players. We didn't want to sell Bale to Real Madrid. Did we accept their first bid? No, we dragged it out and got every single penny that we could from RM. Same with Modric, same with Berbatov and same with Carrick.
Clubs, agents and players shift the goal posts all the time in transfers. Other clubs play silly buggers. i.e. telling players "we're interested in you and will bid. We just need to get rid of some dross first, or we just want to make your club sweat". It's just not simple to go in and just say "here's £20m. Give us Martial."
Levy, Baldini, Mitchell and Poch all know that it would be better to get players earlier and I'm sure they have worked hard to try and get that. If it doesn't' happen, I'm sure it is not for the lack of trying and either because the price being quoted is too high, the club are waiting for a higher offer or the player is waiting for another club. If we take VdV as an example, we would never have signed him under your approach as they didn't want to sell him earlier in the window. If we take Lloris as an example, we would have paid up to £5m more to get him a couple of weeks earlier than we did. That could have been the difference between signing another player in another position that we needed.
Football transfers are a game of poker unless you've got enough chips to **** the other players up. Calling their bluff could work out really well but could also mean game over very quickly!