This already happens. I would be shocked if any mature support branch for a large project __didn't__ have their CRM littered with notes about various companies and specific persons within a given company, even with Consumer Tech Support lines.
Typically, skipping specific steps in common troubleshooting is not done unless the person provides some verification on a new case/call to validate that this step was indeed done, simply because even professionals make mistakes, but specific environmental factors, the person'(s) relative technical knowledge, their behavior in the past with support, common pain points in their environment, and other such things are all recorded pretty heavily.
Usually though this only means that you'll skip Tier 0/1 Support though for most companies. Usually there's always a small reserve of "big guns" that are reserved only for the cases that really need them.
Keep in mind though that this is mostly for companies that specifically look to work with Technology Professionals; when you're talking about generic Consumer Support, such a system is feasible but often non-sensical from any business perspective for the Companies when we're discussing Consumer Support. Comcast's monopoly is only one reason why they were able to weather "worst support ever" for so many years; the simple fact is that they got pretty good at handling vocal customers, or at least adding a public face that made most consumers not care. At best, with such supports, you'll just get X replaced regardless of what X is, and any issue that isn't solved by that tends to be over the head of most people's interests, so any outrage from highly technical persons mostly just gets ignored.
I know what you want, and you can get this with Tech Companies. But if you want fast and good support from Consumer Tech (ISPs, home tech, etc), you have two routes: play the game and just move through the motions ASAP, or post on Twitter and hope that you garner enough attention to escalate your case. Both are a waste of time and energy, IMO, and I know which one would take less of my time personally.