I do not think it matters what type of subject science talks about in the public forum, there will always be those who have issues with some of the subject matter. I think that science should explore all sorts of topics and not have opposing views deter them from discussing them.
Climate change may not be the best subject over which to build bridges because there are so many disagreements about the subject. however, if there were less controversial subjects to be discussed, they may act as an entry point into a larger relationship that provides important cooperation and lasing partnerships.
For better or worse climate change is the most popular scientific topic of the moment. While there are many on either side of the debate, science gains recognition and esteem by discussing this issue. Discussing less controversial but more obscure issues like twin paradox will not increase the esteem of science.
The discussion of less controversial subjects would not give the scientific community more esteem. There are always going to be two sides to an issue, no matter how mundane or clear-cut it may appear to be. And some people will always want to debate a subject just for the sake of debate. While climate change is a touchy subject and probably brings out a plethora of debaters on each side, I would not see the twin paradox as a less controversial topic anyway - it's just more of a speculative idea and not as personal as the climate, which affects everyone.
By favoring more hot button topics, such as climate change, science can remain relevant for the public and increase the reliance that the general public puts into public scientific figures. By only discussing less controversial topics, the science community will become a mere novelty and lose credence. By confronting heavy topics head on, the public will see that the science community isn't afraid of disagreement or being unpopular, which will give science more traction in the public's view.