The idea that a secret identity would be so easy to protect is definitely one of those "less believable bits", and in choosing those words I was trying to be kind.
It's not believable at all. Sure, there are people who will be fooled by that sort of change in appearance, but not enough to protect a secret identity of an extraordinary person. Look at how often famous people get caught trying to go out in public to shop and so forth without being harassed by fans. Then imagine in our modern times trying to be super-someone in secret with cameras, other surveillance and so forth everywhere. I'm sure it was sellable to kids in the 1930's, but today, I feel it's just part of the total set of superhero assumptions we rationalize or ignore away so that we can enjoy the genre's stories.
Because I don't believe people can have super-strength, laser-beam eyes or fly around either; but as a package with a secret identity, I'll let it all slide and enjoy the story.