Back to original question about bridging the gap. I was once told by a wise Rav – Yidden can not pray together, but they can learn together. So, maybe this could be the way – both personally in your interactions and also here. This is a unique place where we can discuss Torah from multiple views and, hopefully, with integrity that Tora requires, rather than approaching issues as politics, where you can quote only sources that support your position and disparage those who hold opposite ones.
Maybe same can be happening at the level of Talmidei Chachamim. It says at the end of Sotah that T’Ch who live in the same town and do not learn together – one dies, another goes to exile… How often do we have such Torah discussions? We hear about Rabbis of previous generation having good personal relationships (R Feinstein/R Soloveichik/L Rebbe, etc), but how often we hear them learning together? Maybe invite a Rav from a different shul to have a public hevrusa with your Rav?