And that's why it's so weird - if he hates them, why does he care about Clara wearing them?

Because it's embarrassing! :D

Archie doesn't mention having living family around any other time, does he?

I kind of feel like he does mention a sister in another book than this, but I don't remember exactly where. In another book he actually mentions his mom coming to visit him and Wolfe, but I'm skeptical of that. (Also in this book Archie makes another reference to Wolfe's mom in Budapest! I thought the earlier mention was the only one.)