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Batman being aggressively secretive towards the Justice League is one of my favorite tropes, but it gets INFINITELY funnier when you take the younger generations of heroes into consideration.
Barry Allen? He doesn't know shit about Batman, still partially convinced he's not human, still a little bit scared to talk to him. But Wally West? That's Nightwing's best friend. He's been around Dick since both of them were kids. He's had sleepovers at the manor since he was thirteen.
I just think it would be hilarious if all of the younger generations 100% know that Batman is Bruce Wayne. They've slept at the manor, some of them have taken various Batkids to school dances, they've been offered Alfred's cooking. And all because they've gotten close to at least one Batkid and said Batkid wore Bruce down until he let them reveal their identity. But all of them manage to keep this information from their mentors. Either purposefully or completely on accident.
Then you end up with a situation where the younger generations grow up and join the League and the older members Very Quickly realize that they seem a lot more comfortable with Batman than they should be.