Thinking about it more, it might be easier to do with a RPi4.  That has 4 pole out that includes audio.  A bit bigger than the zero, but still not huge.

I'm thinking this might be much easier than the Zero.  Here's a video that shows how to do it.

https://www.youtube.com/watch?v=9m8ScViTZi0

Oh, this is a cool idea!   One could do it, albeit not in a totally tiny way, with a Raspberry Pi Zero.   It has composite out.

https://magazine.raspberrypi.com/articles/rca-pi-zero

It wouldn't be a huge box with just a Pi Zero so maybe this would fit the bill?   It could even fit in the TV cabinet for a completely hidden build.

One could also use the Pi Zero with WIFI and expose a webpage to remote admin it and upload video files.