[jamesonquinn]

Sortition (that is, random selection) is a system for roughly-proportional representation; that is, for choosing multiple winners for a legislature so as to represent a population.

Approval voting is a way of picking a single winner.

They solve entirely separate problems.

(Sortition is nearly optimal if your only goal is to match the distribution of voters by a subset of equally-weighted representatives. Technically, it's possible to improve on it slightly by allowing voters to cluster themselves into a hierarchical structure of parties and factions, then performing sortition under the constraint that each party and/or faction must come within 1 seat of its correct proportional representation. This kind of technique is called "reduced-variance sampling" when used in sequential monte carlo sampling.

But sortition, or party-clustered sortition, pays no attention to candidate quality. Most people would agree that if there are two candidates that hold the same ideology, but the only difference is that one of the two has a medical problem that only allows them to work for 1 hour a week, the other one would make a better representative. Sortition cannot be fixed to take this kind of factor into account, which is a big part of why many people prefer more traditional voting methods.)

[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.