My chopper's cooldown is still a lowly 26 (and that's with 2 skill ups at this last MC raid, mind you), but I have all the other guys. I'll give this a go anyway and see how it fares with a max friend MC.
There should be plenty of Maxed Monster Choppers after the clash of clashes. Most people probably got theirs finished or at least reduced their cool down to useable levels.
Max special isn't necessary either, though. Especially if you have maxed GPU. I ran Doffy and maxed him last time using Monster Choppers with high cooldown. It definitely helps, but it's not necessary.