New iPads Won't Connect to Non-Apple Wi-FI

I have had a significant problem trying to connect to non-Apple Wi-Fi routers on the new iPad. I have had 2 different iPad Airs and a Retina iPad mini (all 32GB LTE models), and all have had the same issue. For the first day or so, wi-fi works great. Then it refuses to connect or frequently disconnects from a Wi-Fi network (the indicator icon will often turn off), or even if it appears connected it won't work properly. I've had this issue connecting to a Sprint Sierra Wireless mobile hotspot, and wi-fi in several different hotels.

I've tried just about everything I can think of, including two extended support chats with Apple and a Genius Bar trip. I've tried "Forget this Network," "Renew Lease," "Reset Network Settings,' "Reset All Settings." Yesterday I even erased completely, and rather than install from a backup, I set it up as a new iPad. This worked great for about 12 hours, but then it's been giving me issues again. Since this has happened with 3 different iPads on multiple different non-Apple routers, I'm thinking this is software related. Oddly, throughout all of this my iPhone 5s has no problems connecting to any of these networks, and I never had these issues with the first generation iPad mini. But with the iPad Air and 2nd Generation iPad mini, it's been nothing but trouble connecting to wi-fi.

Note that this does not happen with an Airport router. I can connect at home, or at any Apple store using an Apple router with no issue whatsoever.

Has anyone else had this issue? Any ideas? I really like the new iPads, but can't keep dealing with this. I don't want to be using cellular data all the time.