[eigenket]

It doesn't change the overall point you're making but your "gut feeling" was already wrong 20 years ago. A group working for IBM factored 15 = 3x5 with a quantum computer in 2001.

You are also somewhat wrong about the combining N approximate answers and combining them. It is correct that that the "repetition code" approach (repeat the same calculation a bunch of time and average the results) does not work for quantum computers. However there are quantum error correcting codes which do appear to work, although they require sufficiently small initial error rates (so-called thresholds) in order to help.