It seems like Apple is always tweaking what files will and won’t play natively on the ATV. Apple would much rather you pay them for content instead of ripping media you already own for free.
If you know what you are doing, then aTV is no different from any other player. It's just that it has very limited, albeit quite stable support for formats. AVC video and AAC+AC3 audio have been supported since very first aTV and FrontRow. And only supported container is MP4 (now an ISO standard, btw). Later HEVC video and E-AC3 audio with Dolby Atmos have been added to the list.
To check, what might be wrong with your rips and converted files, use
MediaInfo tool. In general, HandBrake never fails to produce aTV compatible output. Just make sure you use the correct preset and take care of audio settings.
Do bear in mind, that DVD video encoding standard (MPEG-2) predates the H.264/AVC (i.e. Blu-ray) era and can not play natively on aTV. So needs to be converted using HandBrake. And I repeat - it has yet to fail on me, compatibility-wise.
In the end, that's the goal of streaming services... to get you to let them be your library. They are responsible for making sure the formats remain relevant. All you need to do is pay a fee to see them. Eventually convenience wins over thriftiness. It just takes a little time.
That's their wish and vision for blockbusters. I still like to have control over my media library and do not entrust it to someone else's computer/storage system.
And on top of that - I have a library of my own home videos, from my own camera.
I even digitalised movies my father took of me on 8mm film! These things will never be available from any streaming service. So I store them on my own iTunes server. And when I am at it, I store all my other media there as well.
Has worked without problems so far.
One just needs to put some effort in to make sure they are all encoded properly. If you also add metadata, you can organise your media library exactly the same way it looks in iTunes store, TV+ or Plex library for that matter.
If you do not want to bother - take Plex (server) or Infuse (client) or both - and let them try their best in getting the metadata from public sources and handling different codecs and file formats for you.
PS Make sure you have the Web Optimised option checked! It moves the essential movie info to the beginning of MP4 file, so that aTV does not need to skim through the whole file first just to get to the playback metadata (encoding, frame size, framerate etc).