It's a long way from being mostly flat, decisions are made at a crawling pace, and is convoluted enough to hide substantial theft, and shield large numbers of pedophiles .. even those rising to the Vatican C-suite level.
I think it's also possible that says a lot more about the shoddy metaphors of most business schools than the actual level of hierarchy in the Catholic church (beyond the other discussion we were having about how atypical the CC is as an organization).
Well, a thousand years of refinement and a set of company values that most employees have instilled into them since birth will do that to an organization.
That, and also most of the work of a church (interaction with laypeople) is self-evident, local, and easily managed. Once trained, a priest can be set loose on a community without much oversight.
Likewise, I can imagine a chain of Montessori schools being similarly organized, or a trade guild.
Your point is generally good, but not universally true. Mostly-flat organizations do work in certain contexts.