> Does "select a president uniformly at random from the voting population" count as a voting system?

Not voting, but it is a real election system, known as Sortition (https://en.wikipedia.org/wiki/Sortition). It was used in ancient Athens.

Also, yes, approval voting is a great approach and we should absolutely use it. We'd still get people who vote party-line, and it'd take a while for people to truly internalize that they can safely vote for other candidates they like, but it'd be a massive improvement.