Can confirm! My company's typical turnaround time when requesting a PO for non-critical stuff is probably 10-14 days (and trust me, it irritates EVERYONE!), but if someone absolutely has to be paid or something absolutely much be obtained ASAP, we break all our internal rules and can get same day POs. This is obviously not something to be abused, but I wanted to emphasize that you're right.

I encourage everyone to write into their terms what will happen if payments are late. It would be insane not to.

Also, it is a jerkwad thing to do (for some definitions of the term) to right in a destroy clause into your terms (e.g. "When you stop paying maintenance on this software, you must stop using it.") but it really does lock in large enterprises who frequently don't even remember the name of the guy who set the thing up in the first place, and can't imagine committing current resources to figure out how to stop using it.

I feel like I sound either really bitter about my job or that I'm a really awesome enterprise IT manager. :D I'm not sure which is closer to the truth.

This is my strategy for my desktop software, I always make sure the purchasing department and users know ahead of time what is about to happen.