I look at it this way, if I've set aside money for a MAX deal, chances are I'm only going after that one player. So, if I've got the number one offer early on, what's the difference if he signs early or late? If he signs early, great, now I can sit for the rest of FA watching the action from the sidelines. If he signs late, nothing's changed. I'm still certain to sign him and I wouldn't have been making other offers anyway.
That said, has anyone ever had their top MAX offer beaten by another MAX offer (presumably by a team with better coaching, ballpark effects, etc.) midway through, or towards the end of FA?