I'll second the thrift store suggestion. I picked up a Samsung BD-H5100 bluray player at the local FreeGreek for $5 and it has been nice to just pop a disc in and not worry about all the streaming shenanigans.
I'd say you might as well look for a bluray player. Second hand bluray discs are some times cheaper than the DVDs and sometimes the quality bump is nice. IMO, 4k bluray isn't worth it. I've watched a few 4k blurays and while I can tell there's a difference I've never felt myself missing the extra quality when watching a normal bluray.
Another option to consider is an old game console. Anything back to the ps3 has a bluray drive. (Though, not the xbox360, iirc? Also at one point Microsoft forced you to make an account and buy a license to watch blurays, so make sure that's not a thing for any game console you consider.) And I know at least the ps3 had an official remote you could buy so you didn't have to use a controller.
From a privacy perspective, all your options are the same as long as you don't connect whatever you get to the internet.