I've worked for and know many people that work for very large (F500ish) companies, as well as the large consulting companies (Deloitte, Pwc, Accenture, etc), and booking your own flights/hotel is standard practice at all of the consulting companies and most of the others.
Interesting. I’ve never heard of a corporate department that books your travel. Sounds like a hassle. Any company I’ve been at, big or small, just has their policy parameters enforced by third party booking tools.