What fun ... when my folks moved into a senior apartment I inherited my mom’s one-speed coaster bike from about 1950. I remember riding it with her when I was very little (I was in the basket in front!) and remember it being very smooth. I’d like to get it repaired and ride it again, but it’s not a high priority. Would just be fun.
One thing about bike shops — depending upon the shop, they have a lot of great arm-chair mechanics and restoration guys that can take the parts from a 1936 Frigidaire and turn into a unicycle. If you recall the Wright Bros had a bicycle shop/business.
Call a shop and ask if they know someone nearby that could do a rebuild and from there it could get as simple or exotic as you want.