EDIT: For some context, I recently gave podman another go. I have a few services on my homelab server set up in docker containers, so I tried migrating to podman.
After the second major bug (open issue on github) I encountered looked like it would require completely dropping using compose files to work around, I gave up and went back to docker.
I like the idea of podman, but it’s just not stable. I’ll try again in a year or so.
As a bonus, docker’s CLI is significantly nicer.
In that case you can use --log-driver=syslog. There are quite a few other options to choose from :)
Out of interest, why is overlayfs out of the question? If it's because you're running rootless and you're running an older kernel which doesn't allow non-root users to create overlayfs mounts, there's always the fuse-overlayfs driver, which is functionally the same but emulates it in userspace using a FUSE driver. Finally, the vfs driver should ALWAYS work, although it comes at the cost of not being able to share storage between image layers. Also if you're on ZFS there's a zfs driver for that which uses ZFS volumes (same idea as the subvolume approach on btrfs). There's more information on the docs if you're interested, and I'm pretty sure podman has a similar suite of options.