On thing b, before I start talking about the actual common ground search...
A key underlying question is why exactly the people you are appealing to are getting this 'off' feeling; my model of them says this makes sense, but has too many explanations for why they do, some charitable and some not so charitable. In particular, I have the following question: Is this off-feeling due to:
A] Positive Selection via Points (e.g. it doesn't score enough "my people" points)
B] Negative Selection via Checklist (e.g. you didn't endorse ALL of my groups/causes/ideas)
C] Negative Selection via Opposition (e.g. you said something that makes you off)
D] Selection via Alliance (e.g. you are not explicit that you are left-wing and/or fail to condemn the outgroup)
This also points to what one might consider the 'problems in thinking' in the sense that [B] and [D], and depending on details [C], can be considered failure modes.
The hope is that we are dealing with either [A] or at least [C] in a way that can be fixed; the question above implicitly wants to believe [A].
If it is [A], then "all we have to do" is find common ground. This is a hard problem but sounds solvable, and if done right seems compatible with what we want to do.
If it is [C], then our job is to, essentially, censor things until nothing is too weird or too much in conflict with their goals. There is an optimal trade-off here, either moving along the production possibilities frontier or finding pareto improvements, but my guess is that we are close to optimal, as we've put a lot of effort into this and made some big compromises. I am not enthused about making more.
If it is [B] or [D] we are drawing dead - the value we are going for would be destroyed by any solution that worked for these problems.