I wonder if you could form a mathematical model of which districts would be most amenable to a dedicated 3rd party challenger. It seems like you'd want a district which is a "safe seat" (if there is a serious R/D contest in the district, people will want to avoid "throwing away their votes" by voting for a 3rd party). But you'd also want to identify a district where people are dissatisfied with their elected representative anyways. Ideally the elected representative is a significant distance from the median voter.

Maybe a good metric would be the ratio of primary ballots cast to general election ballots cast. If a small number of extremist primary voters are de facto selecting the candidate that will win the general and get the "safe seat", that suggests an opportunity for disruption by being a high profile 3rd party candidate who's just a bit more moderate than candidates the extremists select (more appealing to the median voter).

Another idea is to reach out to a bunch of independent candidates and try to recruit them for your new party to work towards critical mass.

Really the big issue here is not lack of ability but lack of will.

Look at the Tea Party and the DSA; they have enough motivated bases to form parties of their own, but their strategy is to co-opt and purify the existing mainstream parties. Equivalent movements in Spain decided to make their own parties and Podemos and Vox became the next two largest parties in Spanish elections.

If you run a third party that is appealing enough, you can certainly have a go at it. I just have literally never seen, I don't know, the Green Party put up a candidate for local city elections.

It's not like party establishments haven't had candidates lose despite all odds. And Independents have had good streaks at lower offices (e.g. Bernie Sanders, Michael Bloomberg)

Depends where you are: In the recent California governor recall, there were two Green Party candidates and one Libertarian.

Governor is already a bit high-level to really have a chance at breaking through, isn't it? Or am I misunderstanding what you mean?

Was responding to the idea that the Green party doesn't put up candidates for local elections... I think I've seen them on smaller races than Governor as well, but this was a recent example that came to mind easily.

Just curious, but how strong are those campaigns actually? Do they do the door-knocking, the ads, and the voter registration?

Not in my experience.