With AWS we do have a direct line to a Technical Account Manager which we utilize for the rare P1 (Prod down) situations. That gets things moving quickly if it wasn't already moving quickly with Support.
On gold support and P1 there is usually an engineer assigned to the ticket. In many occasions you're talking to the person fixing your problem.
GCE is not perfect and neither is their support. But they do try and even when things go sidewise they take responsibility for it, which didn't happen much with AWS, at least in our case.
Also with P1 issues on GCE most of the time an enginner assigned to the case calls me. I've even spoken directly to the head of google cloud platform via hangouts.