I have the 1st Generation Apple TV and it has been working just fine for years, up until last week (week of May 1, 2017).
Like OP, I suddenly have no iTunes Store content under any of the menus (Movies, TV, Podcasts, Music). Thankfully, the movies, music, and photos that I synced from my iMac remain fully available.
However, getting them to play or pause is nearly impossible because the entire interface has slowed to a crawl. Response to any remote button command takes 2 seconds to a minute or more.
I am guessing Apple made software changes to the iTunes Store that are incompatible with the 1st Gen ATV. I think the ATV is repeatedly attempting and failing to contact Apple, and the continuous timeouts are slowing down the machine. Like what happens to my iPhone Mail app when I lose LTE in a tunnel.
I tried disconnecting the Ethernet and that helped slightly, but didn't completely solve the slowness.
I tried logging out and back in to my iTunes Store account (which took an eternity with my long password) but that did not help. I do have 2-factor authentication turned on for my Apple ID, but I didn't get a login failure message, and the settings screen went from a generic "Log In" to showing my Apple ID as logged in.
Apparently, something like this happened to 1st Gen ATV users in 2014, but it was a brief service interruption that Apple was able to resolve within a few days.
I am hopeful Apple might solve this, but with rumors of a new generation ATV coming out, my speculation is Apple is preparing a new backend server system for that, and the new backend is incompatible with the 1st Gen ATV.
It had to happen someday. Let's face it, this is a 10 year old device that predates the iPhone. Honestly, I'm surprised Apple supported it this long with syncing, rentals, purchases, and even syncing w Disney app purchases. Every time they came out with a new version of iTunes, I expected them to drop 1st Gen ATV support. Apple really went the extra mile to keep this thing alive as their technology has advanced. Talk about return on investment.