connect Library to apple tv

Home Sharing works a bit differently with macOS Catalina or later vs. prior versions, as the sharing of the library is activated in System Preferences, instead of the iTunes application. It shouldn’t affect the Apple TV side of Home Sharing.

System Preferences﹥Sharing﹥☑︎Media Sharing﹥☑︎Home Sharing﹥[set matching Apple ID account and Library name]

The managing of the content of the library for videos is done in the TV app on Mac.

Use Home Sharing to share media from a computer to other devices - Apple Support

Try the following first and see if this helps:

• Unplug Modem from it’s power source for 2-3 minutes and then plug it back in.
• Unplug Router from it’s power source for 2-3 minutes and then plug it back in.
• Unplug Apple TV from it’s power source for 2-3 minutes and then plug it back in
• Restart the computer in Safe Mode (if it's a Mac) and then restart normally and retest the system.

both are on same wifi!

If you have separate SSID network names for the 2.4 GHz and 5 GHz Wi-Fi band on your router/gateway/​access point, then that may lock out devices on each of these to connect to devices on the other part. It is recommended to not separate them, but to bridge both parts into one logical network under one name, and let devices connect however they prefer.

Try restarting the ATV 3 and restarting the computer. Then attempt to re-connect. Frankly I'd suggest updating the ATV to an ATV 4K model.

I can’t access the computer app on Apple TV

Okay, good clue. So what happens when you launch the Computers app on Apple TV?

An additional condition is, that the Apple TV can only see Libraries shared using the same Apple ID account. Verify that they match.

Do you have any other device to test the usability of Home Sharing with?

E.g. Can you access the Home Sharing library on the Mac using an iPhone or iPad or other computer?