Afaik iq correlates highly with g-factor (scientific name for general intelligence). We don't have much better ways to measure it.

Overall, afaik, again, problem solving often needed in our society correlates with this: "general intelligence". When people say: "But there are many ways to approach a problem, many types of intelligence".

This very well may be, but data consistently shows that people having high IQ scores solve different problems better. So, contrary to popular belief, people having higher iq in general have better talent for languages, same with understanding others emotions and using it to your advantage ("emotional intelligence").

As for reframing an issue and solving it in a different way. This may be valid approach (to teach people etc). But IQ is also time-measured. If your new approach does not help you to solve previously unseen problems quickly, it is not noticably increase your intelligence.

Thus, we see consistently that people cannot really prepare for iq tests much. You only get a few points more if you prepare. Same difference as if you are sleep-deprived.

G isn't so much the "scientific name for general intelligence" so much as one explanation for the positive manifold of intelligence. There are others: mutualism and sampling.

I'm not sure you're right about the trainability of tests, either.