What to do when WiFi doesn't work anywhere except, of course, the Apple store?

Alright, here's my situation.

iPhone 5. Yes, WiFi is turned on. My password is correct, and my router at home works for all other connected iPhones. Neither resetting the network settings nor renewing the lease works. Sometimes it connects to the WiFi, but nothing will load so it just defaults back to LTE. It hasn't worked at all except for a few random times, no idea why. I took it to the nearest AT&T store and it didn't work there either, but they referred me to Apple because it's a warranty issue. And, like a miracle, the WiFi works at the Apple store. Of course. The genius tells me to do the steps above and it still does not work.

It has to be something wrong with the phone, right? Or is there some special trick any of you have found? Anything would help me out!

Thank you.