And in the real New York people are obsessed with superheroes even though they're not real. Imagine how obsessed they'd be if those heroes were real and could be spotted out in the wild. I mean sure, you'd probably have some Gothamites / Metropolites who were too cool to care. But, you'd also have obsessed fans staking out the places where these heroes were likely to appear.

And, for Henry Cavil's stunt, people didn't notice him or didn't recognize him. But, he was in Times Square. The place people would stake out to try to find him would be red carpet premieres of things. Similarly, Superman superfans would stake out The Daily Planet. Even if they didn't know he was Clark Kent and worked there, they'd know that he always seemed to be around whenever Lois Lane was in danger.