It's not the default setup and not trivial to do, but Docker has had the ability to run the daemon as a non-root user for a few years. The standard .deb and .rpm packages even include scripts to automate the transition for you on Debian and RedHat Linux variants. See https://docs.docker.com/engine/security/rootless/

The only thing podman gives you that docker itself can't is running without a daemon at all.