Regard whether spreading rationality contributes to EA directly, I do not have evidence in the sense of data, only in the sense of logic. From a probabilistic thinking perspective, getting people to think more rationally, in an evidence-based manner, about their charitable giving would be likely to lead them to give more effectively, and the EA movement is the best outlet for such charitable giving. I agree there is a danger of the kind you describe about social circle signaling, but we can't be sure without actually testing this with experiments and getting actual evidence about what the world looks like.

Furthermore, as Brian Tomasik describes in his essay, getting people to think more rationally would result in people having better lives overall and flourishing, which is the point of the EA movement as a whole.

So spreading rationality would certainly contribute to the outcome desired by the EA movement, of global flourishing and well-being. I'd also say it would be likely to contribute to the EA movement in particular itself, per my first statement above, but that's a matter of experimenting.