good catch. Just fixed the permissions for the file:
https://framagit.org/Obarun/obarun-install-themes/-/commit/93525625f28c3b9533e5a7c28c741e9a8cf11ba3
So, display-manager@ have nothing to do with the installation of the .xsession file. The only one taking care of it is the boot-user@ service which create it if you use the DISPLAY_MANAGER= variable and only if you use it.
With the obarun-install script, i do not use the boot-user@ DM feature to be able to place the sddm and consolekit service at the root tree. In that case the .xsession is never created and so if a lambda user try to enable a another DM it will get a bad behavior( result: "Obarun is a shit distro").
A repeat again on this thread, i know that the user part is not well implemented, it works but not well implemented. The things will change in the future to provide an efficient and stable behavior about user service implementation.