Onyx and Boyue both produce eInk eReaders running Android 4.X (I don't know if I'd characterise that as "ancient"), which would therefore run "NY Times and other periodical apps", that would probably meet all your other requirements. You can find discussion/comparison of them on the Onyx & Android subforums. Current latest models are AfterGlow2 (aka C67, with slightly different software) and T62+ respectively.
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