Thanks for letting me know that it's not tvOS that's causing the problem, that gives me hope! The batteries seem to be working in both aluminum Apple Remotes: when I press one of the "buttons" the 5th Gen Apple TV4K flashes twice quickly, MOST of the time, but does nothing. Whether the light flashes seems at times to depend on the order in which I press the buttons. Nevertheless, I removed the Duracell 2032 batteries, cleaned all the contacts inside the remote and put in a brand new battery, with exactly the same results. I then found two of my older aluminum Apple Remotes that I hadn't used in years and tried them, with the same results after I took out the batteries, cleaned them (one was tarnished) and the contacts inside the Remote and put in brand new 2032s. I even found an old spare Apple Remote White from the original Apple TV, still in its packaging, and tried it -- the battery is still good! Got the same result, two quick flashes on the Apple TV4K when I press any of the buttons, but nothing else happens.
I tried removing the power cord from the Apple TV4K for several hours and restarted it but the result was the same. I get two quick flashes most of the time, sometimes it depends on the order that I press the buttons. Occasionally it seems that when I press "Select" on the aluminum remotes the second flash drags a tiny bit. In any case, last week both aluminum Apple TV remotes were working perfectly, like they have been for over a decade, and now none of them work. When I get a chance I'll plug in my 4th Gen Apple TV and see if the aluminum Apple TV Remotes still work with IT. (I've still got ALL of my Apple TVs, each of which I ordered the minute it was available.) I may have to do a clean install of tvOS 12.4.1 to see whether that fixes the problem (and to see whether all of my folders get restored to their proper positions!).
Does anyone know if the aluminum Apple TV Remotes will work with tvOS 13?