But being rude and abusive to support staff doesn’t help, encourage, or even compel the support staff do their jobs any better or faster. In fact, I’d wager it’s rather the opposite.

I work in IT (not IT support, though) and I’m fortunate enough that none of my business partners are outright abusive. Even so, I still have some that I deprioritize compared to others because working with them is a pain (things like asking for project proposals to solve X problem and never having money to fund them). If someone was actively rude to me when I had fucked up, much less when I was doing a great job, I can guarantee I wouldn’t work any better or faster when it was for them.