Please let me know if there is already an accepted way to do this.
Early in the install process, you’d have a field to type a hostname of a local machine that you’d like to install like. The installer would download an “Install facts” file and install the new machine like the model machine.
The “install facts” file is created at install time. it contains things like timezone, language, percentage of disk space for each partition (to handle disk space of differing sizes) Optional files selected, username/password for root and for first user - anything needed to make the install a two click operation.
Note that this would be a full new install - not a clone of a machine that has been in use for a while.
For Debian there’s Preseed, for Arch there’s archinstall, for a Fedora/RHEL there’s Kickstart, for Alpine there’s setup scripts, for distros with fully manual installs, you could just write a script?
Automating your install is something any sysadmin and mainly any distro developer will quickly reach towards, so it is something almost certain to exist.
Though, if I understand you, you’d want that to be “sourced” from an existing system, yes? I can see the use of that… NixOS is likely the closest to what you want, since you are always defining a full declaration of your system.