I think *my* idealised end goal would be to have an “Insert [Post Kind]” button in the Block Editor, which when I give it a URL inserts a Block with all of the correct mark-up for the information available (essentially inserting an equivalent of the current post_kinds template files). Ideally the block would give me options to toggle icons/snippets, edit some of the data, etc.
By having the Post Kinds as a Block, the user would be free to insert the information wherever they like in the post, and even have multiple “Kinds” attached to a single post.
Admittedly, I don’t know what would be needed to enable this, so I might be imagining something which isn’t realistically possible…