What about Approval, Score, or Condorcet Voting Methods?

Yes, there are a seemingly endless number of alternative voting systems in existence. Unfortunately, none of them is perfect. Indeed, for any system one can find a desirable property that the system does not satisfy. However unlike these other systems, ranked choice voting is tried, tested, and works well in practice. Approval, score, and Condorcet methods are not used anywhere in the world for governmental elections, so whatever benefits they may convey are unproven in the context of real, political elections.

On the theoretical front, approval and score voting fail a critical test that voting theorists call the majority criterion. This is the simple property that a candidate who is the first choice of an absolute majority of voters should always win. RCV, Condorcet methods, and even plurality voting satisfy this, but approval and score do not. Under approval or score, a candidate could be the first choice of 99% of voters and still lose in theory.

RCV is also one of the few methods that satisfy a property called later-no-harm, which we believe is necessary in the context of high-stakes, competitive elections. Namely, under RCV a vote for your second choice cannot hurt the chances your first choice will be elected; a vote for your third choice does not hurt the chances of your first or second choice; and so on. This is not true under approval, score, or Condorcet voting — a vote for a later choice works against an earlier choice. When a voting system violates later-no-harm, voters face pressure to bullet vote, meaning to cast a vote for only one's first choice. Case in point: approval voting was used to elect the Dartmouth Board of Trustees, but once the Trustees elections became mildly competitive, they abandoned approval due in part to concerns over bullet voting. The more voters bullet vote, the closer the system will resemble plurality in practice, and then we're back to square one in terms of improving our voting system.

We also consider the Condorcet criterion to be important. This is the property that the candidate that would win a head-to-head race against every other candidate should always win. While RCV, approval, and score voting may fail to elect the Condorcet candidate, in practice RCV has done so in virtually every single election. Due to strategic vulnerabilities of Condorcet methods, including later-no-harm, and the additional complexity Condorcet requires to resolve cycles, we strongly prefer RCV for political elections.

Furthermore, approval, score, and Condorcet were all designed to be used in single-winner elections only. Ranked choice voting works well for both single-winner and multi-winner elections. For elections that involve a mixture of single-winner and multi-winner races, we strongly prefer the simplicity of using a uniform voting method across the board.

These reasons help explain why ranked choice voting is the preferred voting method of the major electoral reform organizations around the world, including FairVote in the United States and the Electoral Reform Society in the UK. Again, no system is perfect, but if there's one thing all advocates of alternative voting systems agree on it's that our current plurality voting method is surely the worst.

Further reading:
Sightline's Guide to Methods for Electing an Executive Officer
Why Approval Voting is Unworkable in Contested Elections
New Lessons from Problems with Approval Voting in Practice
Single Winner Method Comparison Chart
Why I Prefer IRV to Condorcet