One of the lessons in that book is that the main reasons things in IT are slow isn't because tickets take a long time to complete, but that they spend a long time waiting in a queue. The busier a resource is, the longer the queue gets, eventually leading to ~2% of the ticket's time spent with somebody doing actual work on it. The rest is just the ticket waiting for somebody to get through the backlog, do their part and then push the rest into somebody else's backlog, which is just as long.
I'm surprised FAANGs don't have that part figured out yet.
I do know the feeling you're talking about though, and probably a better balance is somewhere in the middle. Just wanted to add that the solution probably isn't "Let devs deploy their own services without review", just as the solution probably also isn't "Stop devs for 6 months to deploy services they need".
If you had advertised this as a "regular service which happens to use LLM for some specific functions" and the "output is rigorously validated and logged", I am pretty sure you would get a green-light.
This is because their concern is data-privacy and security. Not because they care or the company actually cares, but because fines of non-compliance are quite high and have greater visibility if things go wrong.