It's doing a better job for most of the use cases I had recently.
You can run it rootless which makes it a far better use cases for a lot of dev use cases (where you don't need to emulate a lot of network aspects or heavy filesystem usage).
Also you can run it as root without deamon which is much better for a bunch of use cases and eliminates any problems with fuse-overlayfs. This is the best way to e.g. containerize systemd services and similar (and security wise still better as e.g. it still works well with the audit subsystem).
If worse comes to worse you can run it as a deamon in which case it's compatible with docker compose, and has more or less the same problems.
In my experience besides some integration testing use-cases is either already a good docker replacement without running it as a deamon, or the use case shouldn't be handled by docker either way (sure there are always some exceptions).
Lastly I had far less problems with podman and firewalls then with docker but I haven't looked into why that's the case. (But it seems to be related to me using nftables instead of iptables).