As succinct as I can make it
1) "Intelligence" is a word that has been used in all sorts of ways, but never defined or restricted to one of them
2) it therefore cannot be measured.
3) so what is being measured with IQ tests? Performance on one short test of a set of tasks. Interesting, but not that interesting
4) neither do the tests address innate ability vs practice or preparation at those tasks - surely precisely what they are trying to measure?
5) the predictive power of IQ tests wrt later life success is their main selling point. But predictive power is just a kind of correlation (school success also correlates to later success, but we wouldn't say it measures intellectual talent well) It does not mean that IQ tests measure intelligence, whatever that may be.
6) tests only measure limited problem-solving abilities. Several other abilities based on memory, directionless creativity or emotional thinking are profoundly important.
On this last point I'd say that there are many different styles of thinking with different strengths and a better study of "intelligence" would evaluate these.