Conditions in NY are atypical and the success of the Working Families Party can’t be trivially duplicated but it’s one of the best examples of long term strategic third party activity we have in the U.S.
Most third parties in the US (unlike WFP) tend to go for a strategy for shooting the moon and aiming at top spots immediately, which has never worked. Third parties in other countries generally have a go at wining local seats first and building their base from there.
The US lacks a third party like the UK’s Lib Dems or Greens, or Canada’s NDP. (The US has a Green party that is pretty farcical and doesn’t actually win elections.)
There are usually significantly more than three names on a Presidential general election ballot. The US has a two-party system because it has an electoral system in which voting for any but the least offensive of the two major parties structurually is subiptimal in achieving ones preferences, not because there are only two options on the ballot.
For the local school board or mayor, this is less true, because nearly all third parties do not field separate candidates in these races. It’s possible to win these races since the margin of victory is measured in the hundreds or single-digit thousands, but none of the third parties build up a base and instead shoot for the moon and inevitably lose.
In mich of the country, school board (and in somewhat less, mayoral) elections are nonpartisan. Its true that some local races are poorly contested, but that's actually largely unrelated to party structure, and more related to an insufficiency of people interested in the job (this usually isn’t a problem for mayors and schoolboards.)
Where these races are partisan, minor parties frequently contest them, and even where they aren't, minor parties can be found organizing around them (though where they are formally nonpartisan, you may not notice this, since the endorsement of the third-party may only be actively highlighted to party members.)
> It’s possible to win these races since the margin of victory is measured in the hundreds or single-digit thousands
And third parties do win them. Virtually all of the third party elected officials in the US are exactly this kind of local official, with a smaller number of state officials.
There has basically been no successful attempt to set up an ongoing party infrastructure and base, that actually wins elections. Pretty much all independent politicians in the US are one man shows.
It's really the battle of who can win (then do the same thing for the billionaires) while virtue signaling to their respective "base" of commoners (both left and right). It's all establishment.