Apologies for the patronising comment, but do you really mean that? Docker in Docker works but is intended for the developers of Docker to debug Docker itself. Usually for running Docker from within a container, you just hook up the Docker client to the TCP port of the Docker daemon running outside the container, which isn't strictly Docker-in-Docker.
I ask this in case you're trying a wildly use case (if you really are running true Docker-in-Docker), or are making an unfair comparison (if you're just using the usual Docker client in container to daemon on host). In the latter case, I must admit I don't know what the idiomatic alternative would be for podman, given that I know nothing about it except that it's daemonless (and even that I only learned by reading your comment).