Modern Managed Columns and Content Types is broken or unfinished at best

Ivan54 As you've seen, there's no 'easy' way to get a published content type in a site, though they've made it quite easy to get one into a library.

If you're willing to write some JSON and run some PowerShell, the following should work:

1. Create and publish a new content type in the gallery with ALL of the column definitions that you want to be available on your sites.

2. Create a Site Script JSON file that uses the https://docs.microsoft.com/en-us/sharepoint/dev/declarative-customization/site-design-json-schema#addcontenttypesfromhub verb to add the content type to the site.

3. Use PowerShell to create a Site Template which includes your Site Script

4. Optionally, set the Site Template as a 'default' so that it is automatically applied to all new sites as they are created.

REF: https://docs.microsoft.com/en-us/sharepoint/dev/declarative-customization/get-started-create-site-design