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There are different kinds of political parties. LOCALS sounds like a single-issue fusion party as described here: https://open.lib.umn.edu/americangovernment/chapter/10-6-minor-parties/
Fusion parties choose one of the main two candidates as their candidate. This gets around the spoiler effect. Eg the Populist Party would list whichever of the big candidates supported Free Silver.
A problem with that is that fusion parties are illegal in 48 states(?!) because the major parties don’t want to face a coalition against them.
LOCALS would try to get the democrat and the republican candidate to use Co-Co to choose their policies (offering the candidate support in form of donations or personnel), and if they do then they get an endorsement. I’m still a bit iffy on the difference between an interest group and a political party, so maybe you are in the clear.
https://en.m.wikipedia.org/wiki/Electoral_fusion_in_the_United_States
I love your vision of how a politician should answer the abortion question. Separating the three questions “who do voters think is qualified” “what do voters want” and “what is true” would be great for democracy. Similar to: https://mason.gmu.edu/~rhanson/futarchy.html
When it comes to local vs not local, if 1/100 people is an X, and they are spread out, then their voice doesn’t mean much and the other 99/100 people in their district can push through policies that harm them. If the Xes are in the same district, then they get a say about what happens to them. I used teachers as an example of an X, but it is more general than that. (Though I’m thinking about the persecution of Jews in particular.)