So, why do people care so much about IQ in the first place? If it determines your success, then it doesn't matter what you do, you might as well learn what you are passionate about. If it doesn't, then you should learn what you are passionate about since that'll raise your IQ...
The real issue is that people think success is a determiner of human worth. So, if people are inherently less capable of being successful than others, they think they are inherently less of a human. The IQ debate is really a debate about what determines human worth.
Politics. The political left is the faction of academia. As institutions, the universities and the teachers unions wish to reallocate money to education spending [1]. To do so, they must simultaneously get people to believe two things: 1) that scholastic achievement is extremely important and 2) that everyone can achieve any scholastic goal. Conversely, those on the right wish to debunk #2 so that politicians will stop reallocating money to academia, and instead allocate it to their favored causes.
Scholastic achievement is closely related to tested IQ. The SAT is a not-so-thinly veiled IQ test. Every American is taught by the educational establishment that both scholastic achievement and the SAT are very important to our future. For the first twenty-two years of our life, we get graded continuously on scholastics, but rarely on character. In all, we are raised to be obsessed about IQ, and we are caught in the midst of a political battle where trillions of dollars are at stake.
[1] Note - I do not impute cynical intent to the actual individuals comprising the educational institutions. Institutions promote themselves via selection effects. True believers who believe what's good for Harvard is good for the country will fill the ranks of Harvard and enter leadership positions. Cynics will find jobs elsewhere. Thus an institution will fill with people who sincerely believe in promoting the institution, whether or not the institution actually is doing any good. Thus the actions of an institution as a whole can be self-serving at the harm of the whole, while the individuals comprising it are sincere. Other institutions besides academia have the same effect, see for instance, the 16th Century Catholic Church.
The real issue is that people think success is a determiner of human worth. So, if people are inherently less capable of being successful than others, they think they are inherently less of a human. The IQ debate is really a debate about what determines human worth.