podman is not a drop-in replacement for Docker. You can replace it with podman but expect to encounter minor inconsistencies and some major differences, especially if you use Docker Compose or you want to use podman in rootless mode. It's far from just being a matter of `alias docker=podman`.
The only non-deprecated way of having your Compose services restart automatically is with Quadlet files which are systemd unit files with extra options specific to containers. You need to manually translate your docker-compose.yml into one or more Quadlet files. Documentation for those leaves a lot to be desired too, it's just one huge itemized man page.