This is an interesting question. Before I even attempt a broad answer, I want to note that we do not have professional consensus on the details here. However, I will cite some research that you can maybe check out and see what you think.
Is general intelligence meant to underlie all intelligence?
To answer this, we first have to define what general intelligence is. A good working definition is the following:
g denotes the shared variance in a set of intercorrelated cognitive tasks.
So, when you look at people's scores in a number of cognitive tasks such as arithmetic, vocabulary-based tasks, etc, you find that this performance is highly correlated. Using factor-analysis, we can load these performances onto a general intelligence factor.
We are basically claiming that this high degree of correlation is the result of some underlying factor. (Spearman, 1927; Flanagan, 2014).
Now, it's important to note here that we can't directly test for g. We can only test for other cognitive factors that load onto g.
Test scores as a product of learning and not internal ability
So, you suggested that IQ could be a result of teaching success and that's a problem. I think IQ is a result, at least partly, of teaching success. The IQ test isn't designed to differentiate between biological tendencies and learned abilities.
g is also not necessarily purely biological or learned, it's usually seen as an factor that combines these together. In fact, most cutting edge models suggest that both innate ability to learn and ability that is gained in life load onto g. (Flanagan, 2014).
IQ Scores and g
Now, IQ isn't supposed to be a measure of g. It is rather a measure of a more limited kind of intelligence that combines reasoning, working memory, and a couple other cognitive abilities. (McGill et al., 2018)
It's got its uses, though these are limited and we are probably paying too much attention to it at the moment. IQ test scores are highly sensitive to test-taking conditions, internal and external. (motivation (Duckworth et al., 2011), attention, environmental conditions, etc).
So, while there is a credible theory for general intelligence, IQ probably isn't measuring it. It probably loads onto it, so it may reflect some aspect of g, but it's not directly measuring our general intelligence.
Other Theories and Literature
So, as far as I understand it, the idea of a general intelligence is widely accepted in the literature right now. However, there are detractors, famously Gardner (1983) who proposed that there are different types of intelligence that can't be loaded onto the same one.
While there is some support for this (Visser et al., 2006), I find that the evidence is largely unconvincing at the moment. There is evidence to suggest that these "different" intelligences just load onto g, they are second-stratum. MacCann (2014) does this for emotional intelligence and there's others abound for the rest. (The exception is possibly kinaesthetic intelligence? I don't have that much research on this).
(I would be remiss if I didn't mention that Gardner himself suggests that multiple intelligences can't really be tested empirically since they are only exhibited properly in naturalistic settings. I wouldn't want to hold a position on this matter that isn't empirically supported so I can't defend him.)
Summary
So, long answer short.
- There is (probably) a general intelligence factor
- This general intelligence factors includes stuff that was learned and stuff that's biological, so it does include teaching success
- IQ tests are not testing for the general intelligence factor, it is testing for a subset of cognitive abilities
- In fact, we probably can't test for the general intelligence factor with a single overarching test anyway
- I'd recommend looking into CHC intelligence models for a cutting-edge cognitive model of how intelligence works