The primary downside to approval voting over a good ranked-choice-voting system (IRV is poor, yet is often considered synonymous with RCV), is that it encourages strategic voting.
Let's take the 3 person example and assume I like candidates A, B, C, in that order. With RCV, I would just rank them in that order. With approval voting, I might want to withold my approval for the most popular opponent to A, in order to boost A's chances of winning.
I like approval voting for large democracies because of the reasons you mention and the fact that it is very simple. Trust in the voting system is paramount for a healthy democracy, and requiring algebra knowledge to understand the how your vote is counted is counterproductive for generating such trust.
The main reason "strategic voting" is a big deal when evaluating ranked-choice voting systems is that they actually experience spoiler effects (which more or less follows from Arrow's theorem, no amount of epicycles will totally mitigate them). Approval voting essentially pulls the bottom out of the two most impactful forms of strategic voting by satisfying independence of irrelevant alternatives and structurally enforcing use of the full range.
> Let's take the 3 person example and assume I like candidates A, B, C, in that order. With RCV, I would just rank them in that order. With approval voting, I might want to withold my approval for the most popular opponent to A, in order to boost A's chances of winning.
In a 3-candidate election, the bullet vote for A is not a strategic ballot, it honestly expresses your preference A > B,C. It would be strategic to vote A,C (instead of A,B,C), but it turns out that in elections of up to 3 (relevant) candidates, one of the honest ballots is always optimal.
(This is not to dismiss the impact of voting "for A" vs "against C", because that does matter. But there's a real trade-off there if you aren't fully informed on other voters' (adversarial!) ballots to discourage being the first to actually run that campaign unless you're sure you have a genuine majority.)
Let's take the 3 person example and assume I like candidates A, B, C, in that order. With RCV, I would just rank them in that order. With approval voting, I might want to withold my approval for the most popular opponent to A, in order to boost A's chances of winning.
I like approval voting for large democracies because of the reasons you mention and the fact that it is very simple. Trust in the voting system is paramount for a healthy democracy, and requiring algebra knowledge to understand the how your vote is counted is counterproductive for generating such trust.