[a_imho]

One can most certainly pick even a single winner at random.

Equal chance to represent, there is no fairer way than that.

Voting inherently leads to a money/campaign/popularity contest (and parties, more often than not 2), it is very good at keeping the status quo by feeding back to itself.

I'm not aware of any large scale studies of election method preferences, but keep in mind 1. people are used to voting 2. even if they are aware fairer methods exist their self interest is to keep voting if they think they will benefit from the resulting unfair system.