Not to mention making sure his wings don't bump into things. It's true that normal people can't see our true appearance. At least not without having their cognitions changed. His story of needing the bracelet to hide from us, was frankly pretty weak, but I'm not sure what other lie he could have told. Plus we already knew something was up with him during the school trip when he heard Morgana mentioned pancakes. We just didn't know about him being a crow tengu.

[Yes the pancakes thing happened there too.]

It did explain why he would decline going to the bathhouse across from Leblanc. It would have raised a red flag if he wore just that in the bath, and I guess he didn't want to chance it.

[Or at least he likely guessed Ren would found it suspicious given how Ren once pointed out he was left-handed.]