In either case, whether the container uses a root or user within it isn't really a factor. There's good reasons to assume user permissions within the container, but they have nothing to do with rootless containers: the idea that a regular user can launch a container is a different concept, from the permissions inside the container.
From what I've read, the user running a container really starts to matter when the container is given access to the filesystem (using a volume). It sounds like the user within the container still wouldn't matter in this situation, but the user running the container on the host system would have to have appropriate permissions to the volume.
Honestly, I think the default behavior of rootfull docker is broken by design. Being able to run rootfull docker commands is equivalent to having sudo privileges, because the docker daemon has root privileges and will mount arbitrary files on your behalf.
(I tend to write go and CGO_ENABLED=0, so I typically just pick any user and it doesn't matter. But it can matter in many common cases, so it's worth checking now to save confusion and trouble later. But, it's /etc/passwd that names the user, and filesystem internals that control file access. Typically if a container is designed for nonroot, the relevant files will be owned by the nonroot user they picked. But not always!)