As far as I understand the Conan documentation, a Profile is something that must be managed manually by the user. Is this correct?
It would be neat if profiles and toolchains could be pushed/pulled like regular packages. I understand that the toolchain itself can be pushed as a tool-package (build_requires), but then you still need the profile to fetch and use the toolchain package.
Is there no other option than to simply have a git repository for example, with an organization's profiles, and then install them manually when you need them?
You are looking for Conan config command. Conan config is not only about conan.conf file.
With conan config install, you can import external profiles and install directly to your Conan cache folder.
As example, you can use Bincrafters' profiles available on Github. As you can see, it's a simple git repository with a folder and some profiles.
You can do the same for your company, create an internal git repository and share most regular profiles used by them.