I might have a winner. I have a Acer Aspire One AOA150X netbook from 2008 with a Intel Atom N270 and 1GB DDR2. Here is an Imgur album of my experiments with it in 2018 plus a picture from today.
I don't have all of the details because this was years ago. The benchmarks were within 1-3 FPS, but in practice it didn't matter because the actual experience was seconds per frame. When staring at a wall on itemtest it showed signs of life by pushing hundreds of FPS.
I used a stripped-down sketchy version of Windows 7. I definitely must have known at the time that Linux would theoretically be way better for this sort of thing but I must have had my reasons for not using it. If I were to take a guess it would probably have something to do with Windows being net better overall for the ancient integrated graphics by using DirectX instead of OpenGL. Or just that the OpenGL Linux version wouldn't launch at all.
I think mastercoms told me to disable multicore rendering or something equally unheard of. I tried some other software as well.
Team Fortress Classic: Not much better. It could simply be that running an operating system and a 3D application at the same time is simply too much to ask for the poor single-core 3W CPU.
Retail Half-Life 2: I recall this being a single-digit FPS but technically a moving picture unlike TF2. Wall-stares also technically gave me smooth motion.
DUSK: Despite the pseudo-N64 graphics it is in fact a game running on a modern game engine and therefore unusable. It did launch though.
NES emulation: To give no credit at all, I was surprised to see an NES game run at full speed. Mario still wasn't any fun because of the keyboard ghosting.
Steam dropped support for 32-bit hardware so I'm not sure if I could revisit this experiment if I wanted to, but if anyone has any further optimizations I would be down to try to break the 4FPS barrier.