Q: Airplay won't connect

Hi, Baskervillain.  

Thanks for the question and let's see what we can do to get AirPlay working again.  Looking through what you have already tried, I only have a couple more suggestions.  I would like to make sure that all firewalls and any installed security software have been disabled.  

OS X: About the application firewall

Resolve issues between iTunes and third-party security software

If AirPlay still give your the connection error, let's make sure to rule out any network interference, that your router settings have been optimized and that the ports related to the Apple TV have been allowed in your router.  You may also want to make sure that you aren't connected to a guest network and test changing between the 2.4 and 5 GHz options that most routers offer if available.

If AirPlay disconnects by itself, your wireless network might be too busy or have too much wireless interference to maintain the AirPlay connection. Learn how to troubleshoot Wi-Fi connectivity issues. The manual of your network router might have additional suggestions for maintaining a strong and fast connection with your devices.

Use AirPlay to display video from your Mac on an HDTV

Recommended settings for Wi-Fi routers and access points

TCP and UDP ports used by Apple software products

The next step would be testing AirPlay via iTunes in a new user and also in Safe Mode.  

How to test an issue in another user account on your Mac

OS X El Capitan: Start up in safe mode

Please let me know the outcome of all of these troubleshooting steps.  Thank you for visiting Apple Support Communities.  

Cheers