I did it once, I'm pretty much sure Tim did not read the email, why would he, someone in his team did. I had an awful experience with AASP, no computer, no fix, no timelines (and money was tight), tried to escalate as much as I could and failed. Wrote to Tim Cook, explaining my situation and attaching all references... Got a call from Apple within like 2h, got a brand new Mac in a day. Those emails do work.