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This is indeed my problem. And I will pay to solve it, eg. In the form of Plex or Netflix

Netflix/Prime/etc aren't even comparable to a local media library to me. Legal issues aside, a local library can be configured to download the latest episodes of programs I like automatically, and keep an eye out for movies I like the look of, which I can then watch offline at my leisure. Streaming services allows me to browse programs I haven't considered adding to my library, or maybe haven't even heard of, and watch them immediately without having to wait for them to download. They're different use cases for different purposes, and with a few exceptions for hard-to-find stuff that I mostly keep around for nostalgic purposes, streaming is winning. I can't store the programs I want to watch for the cost of a Netflix subscription (a quick calculation shows that it would cost £10 per month in disks alone to fill a 4-bay NAS with 8TB disks, without considering Usenet subscriptions and electricity costs)

Plex has an ATV app

I know. This does not.