You don’t get an option at the bottom that says “upgrade to iPadOS 16“? I’m on iPadOS 15.6 on my Air 5 and I see 15.7.1 but below it says “Also available: update to iPadOS 16”. It just says “up to date”? I’m not entirely sure how that works as I don’t update, ever, and I’m not on iPadOS 15.7.1. Otherwise, try iTunes on a computer.