Yep. I worked tech support part time in college- I remember. :)
When implementing an "everyone does tech support" system, though, remember that at least some of your employees won't have much trouble leaving if they find themselves hating the experience. When they have to deal with customer complaints, but can't do anything to fix them, they may well move to a company where that's not the case.
You have summed the standard customer support experience so distinctly.