I think power *can* make some difference, but you do hit diminishing returns.
Obviously the more convoluted the air path, the more power you need to get the same thing done, but there are limits. Does a 1200W Sebo D4 perform that much better than an 865W Filter Queen 95x? Not really, and most of that difference is the power nozzle (Sebo's is really good, FQ's is merely OK). For uprights, I can pretty confidently say that my Kirby Heritage II Legend outcleans my Royal 880 (4.5A motor on the Kirby vs 3A on the Royal). Both of them keep up with a 12A Simplicity 6970 fairly well, too. I'd even say the Kirby does better.
As another example, consider the Hoover Concept One. Only a 4A motor *and* it's a clean-air design - but it's widely regarded as being just as good a deep cleaner as Sanitaire, Kirby, Royal or Simplicity.
On the other hand, some of these wickedly overpowered 2600W canisters (like the Hoover I had when I lived in Italy)? They don't perform any better than a 1000W Tri-Star, really.