Premise 3 of your argument is strictly false. Mathematicians who are not heavily invested in ZFC or even who are but who are as yet unfamiliar with Easton's theorem tend to 'have an intuition' that ℘(A) > ℘(B) whenever A > B. It is intuitive in the quasi-perceptual, not the "I have a hunch," sense, that 2n > 2m when n > m, because we can write out both powersets directly and count their elements, and it is not a hard task to perform transfinite induction over the finite n and establish the relevant principle.
But vs. infinite sets and forcing, it is possible to have 2ℵ0 = 2ℵ1, and these can both be equal to 2ℵ2, 2ℵ3 ... 2ℵω ... etc. Moreover, unlike the finite n, ℘(A) for some transfinite A is not only equal to 2A, but 3A, 4A, ... up to AA. By contrast, there are only two cases among the finite n where the basic tetration set over a number is equal to the powerset of that number, viz. for 00 (if you accept the arguments for 00 = 1, btw!) and 22. But 2 by 2 is 4 regardless of whether "by" means addition, multiplication, exponents, tetration, etc. So whatever intuition tells us about finite powersets, it seems ill-equipped to tell us as much about infinite powersets.
Objection: humans do apparently share a range of perceptual types: everyone is at least able to see or hear or taste or ... And if we believe in not Platonic, but Kantian forms†, i.e. space and time as formal intuitions, then if our spatiotemporal perception is continuous, then one wonders whether we have something like an intuition of "the" Continuum, so that we can vaguely quasi-perceptually discern that it is not equal to the set of all countable ordinals. It took a long time for people to come up with and accept the notion of different-sized infinities, though, and I confess that had I not had Cantor's zig-zag argument shown to me, I would have been tempted to think that there are more fractions than integers; so my ultra-naive set-theoretic self would have gotten a fairly simple matter quite wrong, on grounds of my deference to my preliminary intuitions.
At best, and only if I knew that the set of ratios of integers was equal in cardinality to the set of integers, but not the set of real numbers, I might've thought, "Hmm, well, we learn about the irrational numbers based on learning about the set of integer ratios, so it's like the set of irrational numbers is the 'conceptual successor' of the other set, so..." in which case my intuition would have favored CH, though.
But so if evolution had instilled ~CH in our minds, then why did it do so? And how further specified an installation are we speaking of? Did it also implant, "2ℵ0 = ℵ192,315," in anyone instead? Or any other specific such equation? I would suspect that evolution, if it had a "direct hand" in such matters, would still have impinged more on our "sense" of GCH rather than CH. Yet once again, and historically, GCH did not "intuitively occur" immediately to Cantor and others during that initial period, even if CH did, and on top of that Peirce came up with an incorrect argument for the nonexistence of ℶω (though it also came to be known that something similar to Peirce's argument was true, namely that ℶω, or even ℵω, is effectively a 'large cardinal' vs. ZFC minus the replacement scheme).
So you might be asking for an explanation of a nonexistent fact, and it is not clear enough whether you are talking about even a possible fact (aside from, perhaps, bare 'logical' possibility, but almost certainly not, it seems, nomological possibility) that we can at least intelligibly imagine counterfactual explanations of. And I don't know what an analytical (or continental!) philosopher (as opposed to a set theorist or maybe a physicist with a penchant for neurobiology or cognitive science) would be able to do to explain why evolution 'told' us ~CH, even if it had told us this.
† From Britannica: "For the 18th-century German philosopher Immanuel Kant, form was a property of mind; he held that form is derived from experience, or, in other words, that it is imposed by the individual on the material object. In his Kritik der reinen Vernunft (1781, 1787; Critique of Pure Reason) Kant identified space and time as the two forms of sensibility, reasoning that, though humans do not experience space and time as such, they cannot experience anything except in space and time. Kant further delimited 12 basic categories that act as structural elements for human understanding."
EDIT: To further illustrate the non-intuitive/counterintuitive qualities of transfinite arithmetic, I will bring up the difference between
ℵ0 + ℵ1 + ℵ2 + ...
ℵ0 × ℵ1 × ℵ2 × ...
König's (set-theoretic) theorem is, vs. its simplest case, that the sum of all the ℵn is smaller than the Cartesian product of all the ℵn. Therefore ℵωℵ0 is greater than ℵω. By contrast, for 𝔠 = the cardinality of the Continuum, 𝔠ℵ0 = 𝔠. This entire phenomenon (singular cardinals admitting of a different transfinite arithmetic than regular cardinals) is hardly 'intuitive' in the required sense of 'intuitive' (AFAIK König incorrectly concluded that the Continuum didn't have a cardinality assigned to it among the alephs, to Cantor's renewed consternation at the time).