In order to save others the mind-boggling level of frustration that can be achieved when attempting to troubleshoot Home Sharing between iTunes and your AppleTV, I want to highlight an item from Step 4 of the Apple support troubleshooting page:
Your devices are not using a Virtual Private Network (VPN), or that they are all on the same VPN. A VPN may isolate the device and cause connectivity disruptions.
That sentence does not tell the whole story. Not only should your computers that have iTunes on them not be running a VPN connection, but there should not be any device on your network that is connected to a VPN, even if that device has no iTunes installation.
I figured this out the other day when I hastily read the support article and overlooked this step as the computer with iTunes Home Sharing was not running a VPN. However, my work computer was on the network and actively connected to our company VPN.
After giving up on Home Sharing, I went into the office, did some work, and shut down my VPN connection. ‘Lo and behold! Home Sharing is suddenly working again. I hope this helps at least one other person because it took me a couple weeks — and the mistaken assumption that a recent AppleTV update broke the Home Sharing — to figure out what was really happening.