the thing with uncommon beliefs is not that they are likely wrong. but that digging in your heels is surely going to fail, regardless of who is actually right.

so your suggested response is the right approach, but it doesn't end there. you can try find a common belief and build up your argument from there. peoples opinions can be changed if you take the time to learn how their opinions are formed and present them with the opportunity to consider alternative ideas. ideally in such a way that they discover the truth on their own.

a key component is that unity enables change. it is better to be wrong but united, than right and divided. if we are united (and thus stay friends) then we can learn from being wrong and change direction. if we are divided then changing direction is difficult.

I think there are also different definitions of "digging in your heels".

What most people do is just whine and repeat themselves because they don't understand all the ways they're being misunderstood. They lack self-awareness because they lack sufficient experience hearing and digesting the arguments from the other sides. This is a missed opportunity.

What people should do instead is leverage their self-awareness once they have the spotlight and "magically know" which concerns to address when they are given that brief window of rebuttal. It's hard to get attention, so they must strike when the iron is hot. It takes a lot of experience, and most never get to that level. Repetition signals to everyone else they don't really know what they're talking about.

The majority of the audience may actually be on your side agreeing with you, but they won't stick their neck out for the truth if they know they're less informed and less experienced than you, yet even you still failed. They have no chance to do any better, so they just shut the fuck up. Everyone languishes. Your point is noted, but not winning. All you did was paint a target on your back for the next time you say anything. People would rather be winning than right. Agreeing with you once doesn't mean they side with you.