> What I hate about docker and other such solutions is that I cannot install it as nonroot user

There's a rootless [0] option, but that does require some sysadmin setup on the host to make it possible. That's a Linux kernel limitation on all container tooling, not a limitation of Docker.

> and that it keeps images between users in a database.

Not a traditional database, but content addressable filesystem layers, commonly mounted as an overlay filesystem. Each of those layers are read-only and reusable between multiple images, allowing faster updates (when only a few layers change), and conserving disk space (when multiple images share a common base image).

> I want to move things around using mv and cp, and not have another management layer that I need to be aware of and that can end up in an inconsistent state.

You can mount volumes from the host into a container, though this is often an anti-pattern. What you don't want to do is modify the image layers directly, since they are shared between images. That introduces a lot of security issues.

[0]: https://docs.docker.com/engine/security/rootless/

If I install podman on my Linux machine, it's rootless by default. No fiddling required of me.

Docker could do a lot better job in the packaging of their software. Even major updates require manual uninstalling and reinstalling it... Podman just works.