WiFi has no internet access...But it does! I'm writing this question while connected to a friend's WiFi, with mobile data turned off. But my phone says "connected, no internet access", and there's a ! by the WiFi indicator on the bar along the top of the screen.

I think android thinking there's no internet is stopping certain apps from working, where as others don't seem to care (and work).

Why is android thinking there's no internet when there is? Can I assure it that there is internet?! I'm on 6.0.1
"Connected, no internet access" means that the test that Android uses to check if there is Internet access is failing. The url or method used to test if Internet is working varies by Android version and on 6.0 it uses connectivitycheck.gstatic.com/generate_204 as the test. - Reference

"Internet" is just a network, if the routing, access, DNS resolution, etc. pieces are all there, the "Internet" is going to work... regardless if the test Android does to see if it is working fails or not, although you may have to specifically tell your device (via a pop-up toast prompt or setting in WiFi - Advanced) to stay connected to that WiFi network.

Why the test fails? Hard to say, but something in your friends network or their ISP is blocking the check your device makes to test if the Internet is working.