Batman then took the bit and ran with it -- creating counter measures for the others behind their backs and without their knowledge.
Bats probably would have been forgiven if he'd just stood up at a meeting one day and said "Look, we've all seen various members get possessed, mind controlled, what have you. I want to create counter measures that could stop each of you if that sort of thing happened again."
There would probably be some debate and some back and forth but in the end the other members probably would have agreed. And they wouldn't have had to have known the SPECIFIC counter measures.
But instead he did it all behind their backs making it appear to them as if he didn't trust them.
And if you carry it too far you end end up with the whole "Batman is a dick" problem we had in the lead-up to "Infinite Crisis".
In short there needs to be balance.
Tony Stark can be the fun-loving playboy... to a certain extent. Carry it too far and you end up with a character no one takes seriously.
Peter Parker can be a 'loveable loser' but if all he does is 'lose' and he never has any good things happen then he becomes that emo kid that no one likes to hang around with.
Batman can be the control freak with a plan for everything in his pocket but make him too much of that and you end up with 'Bat-dick'.
Superman can be supremely powerful but still be human. He can stand for anyone coming from another culture or coming from a mixed culture background -- how to not lose sight of your heritage while at the same time embracing the society you grew up in. Balancing job and home life and not becoming a workaholic. But if you take any one of those things too far -- make him too powerful, make him the perfect human, then he becomes as interesting as wallpaper paste.