To be fair, this doesn't actually contradict most of the rest of what you say. But this correlation does suggest that there are some shared factors (whether innate, or developed, or both) that affect many or all kinds of "practical intelligence"; one might reasonably call these factors "general intelligence".
Except that this is literally what happens; this correlation between seemingly unrelated cognitive tests is referred to as "the g-factor". https://en.wikipedia.org/wiki/G_factor_(psychometrics)
To be fair, this doesn't actually contradict most of the rest of what you say. But this correlation does suggest that there are some shared factors (whether innate, or developed, or both) that affect many or all kinds of "practical intelligence"; one might reasonably call these factors "general intelligence".