There are two separate issues here that deserve separate discussions:
1. The class of travel an employer pays for or should pay for. FT has a bazillion threads on this already. I have my opinions, which have been posted on FT more than once; I don't want to belabor them here.
2. How corporations book travel. The large corporation I once worked for let us book our own; it didn't have a corporate travel department or agency. I suspect it lost a lot of money that way. Then, when I headed my own company, IBM was one of our clients. We were allowed to use their corporate travel program even when we weren't traveling on their behalf. We'd call their agency, they'd book any tickets we asked for and charge the traveler's credit card. The cost went on that employee's expense report, where any review took place. Then IBM changed its policy, but we saved a lot of money while it lasted. We were too small to get much of a price break on our own, but we were in the airline small business programs like AA's BusinessExtrAA. I don't recall getting much benefit from them.