Abstract thinking is the definition of intelligence because of its general applicability to learning. All learning, be it math, language, engineering, music, etc, amounts to recognizing patterns, generalizing them, and applying the generalization to a different problem.
I'd like to point out that basically every academic admission test (SAT, ACT, GRE, MCAT, etc) is an IQ test mixed in with some domain-specific knowledge testing.
No offense, but having the opinion that intelligence is not measurable is roughly as ignorant as believing the the solar system is geocentric. It requires throwing aside or ignoring a massive amount of scientific evidence and consensus.
Some of the smartest people I know wouldn't do well on an IQ test. Many of the questions are abstract but not all of them -- at least in my case several of those questions seemed more cultural than anything else. (And in some cases these supposedly 'abstract' questions seem to have a possibility-space of answers that even the test-writers did not anticipate.) If you define intelligence as the quantification of one's experiences, then sure, I'll roll with your argument, but experience (and abstract thinking) are just small parts of the overall equation, to which I think we still cannot grasp.
Essentially: predictive power. The reason heliocentricism became the standard model was because it predicted the movements of planets better than the geocentric model.
It's the same with IQ tests. They predict the success of individuals in many different fields. It's a predictor of income, academic achievement, and social status.
What I'm not saying is that IQ tests are perfect, they are not. As you say, there are language issues. But even with those issues, IQ can explain something like 25% of the variance in grades and 16% of the variance in income. It's also telling that more rigorous fields of study (math, physics, medicine) also correlated with higher IQs.
I'll close with the fact that IQ is not everything. It's only a measure of how quickly you'll pick up on something. It also has no impact on a person's mental health or general interests. A lazy person with a high IQ will probably not get very far in the world, neither will one that suffers from crippling depression or anxiety.
These factors seriously decrease the quality of IQ's predictive power, IMO. If you need to attach asterisks to an IQ assessment for a weakly-performing individual ("high IQ but lazy") then I wonder if IQ is the right measurement for that kind of criteria. Yes there is correlation between high IQ and hardcore nerdy professions but that does not mean the two are conclusively, 1.0-correlation linked.
The practice tests I've seen tended to ask a lot of abstract questions that, in some cases, seemed to rely on having a specific set of cultural experiences. I remember encountering a question that asked me to clarify the meaning of a 'common' expression, and thinking, "how lucky am I to have been exposed to and know just what this question refers to? Would I have a lower IQ just because I was an immigrant or something?" (It was the US version of the MENSA practice exam on their website circa several years ago)
I'm of the opinion that intelligence is unmeasurable, if even comprehendable by us mortals, and that the best proxy we have is one's propensity to learn and be curious.
Seems that measuring intelligence is yet another example of the folly of human artifice