I don't know why you're so shocked by this? This is all very basic use of the OCI spec.
> just trying to execute the command within the child container's filesystem
There is no such thing as changing your root filesystem or your uid from an an unprivileged process. There is no "just do it in user mode" either.
So they must have a root process somewhere doing it for you.
I just started googling, apparently for podman the setuid root binary is newuidmap
It's only in recent years that it's become trendy for distros to disable unprivileged user namespaces in the name of reducing kernel attack surface. So while a privileged daemon is needed for those distros, there is no such inherent restirction on the kernel level.
[0] https://man7.org/linux/man-pages/man7/user_namespaces.7.html