I have 3 ATV3s in my home and YouTube has been working fine during the last 6 months(or more).
Have you tried the following?
(1) reset cable or DSL modem
(2) reboot router
(3) change ATV3's manual DNS setting to 8.8.8.8(Google DNS)
(4) restore ATV3 to factory default
(5) sign out of YouTube, or sign in with a different YouTube/Google acct
