Isn't there support for rootless docker these days as well?
I never used it, but have seen separate "rootless" images many times. Seems like you have to build a separate image for rootless Docker? podman runs regular images under an unprivileged user just fine.
You don't need a separate image. You can run a script to install a rootless systemd service that starts the Docker daemon. You just point the Docker client to that rootless daemon's socket and do things as before.
You'll need some configuration if you want to expose system ports (below 1024).