This situation is completely ridiculous. I have had this problem and I have been waiting them to fix it since iOS 8.0 or maybe even before that.
I cannot answer calls, I cannot finish calls, I cannot mute the microphone, put video on... Literally the Facetime app is just the crappiest app my iPad has.
I already restarted the iPad, restore the settings, did everything I could and the problem remains there. It is a software issue and no-one from Apple seems interested in solving it. I contacted Apple support and all they told me was to reset my settings again. Do you guys really think I will lose my time contacting technical support if the problem I had was so simple to solve? I am not one of those '...' people who buy apple stuff just to look cool, I have had this products since some years ago and I regret the way iOS has changed from a 'perfect' operative system where the things that crashed were third party apps to a point where the third party apps actually perform better than the integrated iOS apps.
I bought my parents and iPad, and they have the same problem. Sometimes I am the one who cannot answer calls, sometimes they are the ones who cannot answer the call.
Restarting the iPad SOMETIMES just fixes the issue for a couple days or even just hours.