Could you say more about why you think real analysis specifically is good for this kind of general skill? I have pretty serious doubts that analysis is the right way to go, and I'd (wildly) guess that there would be significant benefits from teaching/learning discrete mathematics in place of calculus. Combinatorics, probability, algorithms; even logic, topology, and algebra.

To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems. I'm not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum. With, say, combinatorics or algorithms, everything is very close to intuitive concepts like finite collections of physical objects; I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.