On the other hand, it would be worth asking how many (if any) great mathematicians had high but non-exceptional intelligence

Some of my candidates (who, perhaps not coincidentally, also happen to be among my "favorite" old-time mathematicians, in the sense of stylistic identification):

• Hilbert
• Weierstrass
• Lie
• Cantor
• Noether

All of these violate (what I think of as) the "math genius" stereotype in some way. None of these were considered child prodigies; in many cases they took up mathematics relatively late (Lie), had some competing interest (Cantor), or stood in contrast to a prodigy they knew (Hilbert, the prodigy being Minkowski).

Expanding the scope to physicists (and in the category of "widely held cultural beliefs that are probably wrong"), I will also nominate:

• Einstein

whom I suspect of possessing significantly less Tao-style ability, and being more akin to the above-listed mathematicians, than is commonly assumed.