While it could get way too meta to let us switch characters after we see what our opposition has choosen, one thing might be to let both sides pick their perks after the teams are shown.
Monster is naturally stronger, it should be. Let us pick hunters after the monster pick.
If not, @Applecrow idea sounds great too.
The blind picking keeps things fresh, it leads people to create choices based on their own playing style and stops cookie cutter counter picking from occurring. Games like this are always going to struggle with balancing, as such, there are bound to be hunter combos that are strong against certain monsters, on certain maps. I agree with their method of concerning yourself with your own personal strategy and leaving the rest to chance.
I completely agree with you. Plus I feel like your own hunter team influences what perk you take rather than the monster.