iPhone 6s incoming calls won't start when Apple headphones are connected For a week or so, my iPhone 6s has been unable to start the call properly when I get an incoming call, and the headphones (Apple EarPods, wired) are connected.

I accept the call, and the view transfers to the "call screen", but the call is indefinitely stuck at the call length time 00:00 and nothing can be heard or spoken.

This problem occurred every time when the headphones were connected, regardless of having restarted the iPhone prior.

When accepting calls without the headphones connected, things work as intended. The headphones can then be connected in the midst of the conversation and work fine.

I'm guessing I need to reset my phone. But since I would immediately have to recreate my settings from an iTunes backup – would it actually solve any problems?
You absolutely will want to do some tests or work with AppleCare / support on this. I get the same very seldom, and if yours happens each / every time your diagnostic logs could let Apple fix this for everyone (assuming there are bugs in the code on answering calls). 

Almost certainly not battery, but it could be. 

I would make a full backup then try in order:


Reset network settings and test. 
Make final backup then don't backup next steps. 
Erase phone, set up as new and test. 
Erase again, restore final backup and test. 
Don't forget to set up backups again.