OP w5jck is right.
Basically, source file is the same no matter how far the technology advances. So, for example, if Apple surprised us with an AppleTV 8K or AppleTV 16K, if the source file is 1080p it is still delivering 1080p on either of those hypothetical AppleTVs too. If you had a SD video from a DVD, VHS or similar scan, even an AppleTV 32K could only play whatever the quality of the source file can deliver- it doesn't get better with hardware upgrades.
For your purposes, a newer AppleTV model will simply be slightly faster hardware perhaps with a few features available to the new ones that isn't available on the HD (if there are any-I can't remember).
Now, an AppleTV 4K can substitute 4K video (streams) in place of 1080p if the file has a 4K version and you purchased it from (and stream it from) iTunes store. If so, that would likely be a noticeable improvement on your 4K TV.
However, your posts read to me like you need to recheck AppleTV and maybe iTunes Store settings. You should be able to notice a quality upgrade difference when pushing a 1080p file to your 4K TV. I'm suspecting that Netflix and Amazon apps ARE throwing 1080p to your TV but wonder if iTunes is doing the same. I wonder if maybe you have an iTunes setting locked on SD and that's why it doesn't look as (relatively) good to you. Look in preferences, playback, download options and see what you have checked there...
I'm wondering if you perhaps have "Most Compatible SD (480p)" checked instead of as you see with my own settings. If so, that would explain your description of inferior picture vs. Netflix and Amazon app video.