I don’t know that the current crop of immutable distros would be able to prevent something like this. rpm-ostree, at least, lets you install out-of-tree rpm packages to the base system, you just have to reboot for them to take effect.

Not really. Immutability can be overriden by root, who can then edit files.

And in addition to that, /etc/, system config files, including pam files mentioned here, are not immuable even in immutable distros.