I'm a fan of AV and pushed for it in the UK referendum, but I'm much less convinced it would work in the US.
In the UK it was a direct counter to the form of tactical voting that already existed. That is to say, people in the UK would often vote to keep a particular party out based on their perceived notions of the others. This lead to absurdities like people voting Labour when their preferred Lib Dems were actually in second place, closer to Tories they so desperately wanted to beat (I grew up in a constituency this happened in).
The problem in the US is more about building third parties. Without existing parties that people actually prefer over the Big Two you aren't going to get much benefit from AV, and frankly it's flaws are more likely to cause problems. There are three big ones I see.
1) Insufficient candidates and hard partisanship can defeat the purpose.
AV is based on the winning candidate being preferred by the majority of the population more than the other options. This means that, ideally, no candidate would win with less than 50% of the vote. Usually when you have a large enough number of parties representing a continuum or at least range of views this is how it plays out; people are more willing to rank parties that they might not otherwise vote for to show preference over the ones they truly despise, so the preferences trickle down till someone gains the consent of a majority.
But in cases like the
Burlington mayoral race in 2006 there simply weren't enough candidates and the vote was too partisan. A full 10% of voters registered only a first preference for the immediately eliminated Republican, leaving Bob Kiss to win with only 48% of the voters registering a preference for him. And this was in a state and election with a strong third party (who won).
"Only" 48% sounds weird compared to a FPTP system, but AV gains a hefty chunk of it's moral justification from actually resulting in majorities, and situations where pluralities can win introduce other errors.
2) Counting orders can make a difference.
This is especially true in heavily partisan races where people don't register second or lower choices and the overall vote depends on a plurality. Some counting systems eliminate all candidates under some threshold (or even everyone other than the two largest parties) immediately while others eliminate them in order from lowest to highest from the first elimination to the winner. In some situations this can change the result of the vote, the former more quickly giving lower preference votes to the biggest parties while the latter allows a consensus choice to build from smaller parties.
3) Weakens mandates.
Especially in cases where either of the above apply and so the AV system itself has given a weaker mandate than it usually does. Especially in the early years you can expect a lot of attacking winning candidates based on their winning on second or lower preference votes. In my view this is part of the point; you have to fight to remain attractive to a wider swath of the population than previously. But in many cases AV could make it harder for such representatives to claim the authority to push their agenda. That can hurt the population that elected them more than having a strong representative who won with a bare plurality in FPTP.