Is "points of normed IQ" the right thing to measure?

No, usually in these cases you will be using an effect size like Cohen's d: expressing the difference in standard deviations (on the raw score) between the two groups. You can convert it back to IQ points if you want; if you discover a d of 1.0, that's boosting scores by 1 standard deviation which is usually defined as something like 15 IQ points, and so on.

So if you have your standard paradigmatic experiment (an equal number of controls and experimentals, the two groups having exactly the same beginning mean IQ and standard deviation of the scores), you'd do your intervention, do a retest of IQ, and your effect size would be 'IQ(bigger) - IQ(smaller) / standard deviation of experimentals & controls'. Some of the things that these approaches do:

1. test-retest concerns disappear, because you're not looking at the difference between the first test and the second test within groups, but just the difference in the second test between groups. Did the practice effect give them all 1 point, 5 points, 10 points? Doesn't matter, as long as it applies to both groups equally and their pre-tests were also equal. The first test is there to make sure you aren't accidentally picking a group of geniuses and a group of dunces and that the two groups started off equivalent. (Fun fact: the single strongest effect in my n-back meta-analysis is from when a group on the pre-test answered like 4 questions more than any of the others; even though their score dropped on the post-test, because the assumption that the groups were equivalent is built into the meta-analysis, they still look like n-back had an effect size of like d=3 or something crazy like that.)
2. you're not converting to IQ points, but using the raw score. This avoids the discreteness issue (suppose the test has 10 questions on it. What does it then mean to convert scores on it to its normed range of 70-130 IQ or whatever? getting even a single additional question right is worth 10 points!)
3. you avoid the issues of IQ points being 'worth' different amounts at different parts of the range. Suppose you took a bunch of IQ 130 kids and did something to boost their scores by 5 points. Is this easier, as hard, or harder than taking a bunch of IQ 100 kids and boosting them 5 points? If there's any differences, we might expect to see them reflected in the standard deviation being larger or narrower, and so this'll be reflected in our effect size.

Effect sizes are also the sine qua non of meta-analyses, so by thinking in effect sizes, you can more easily run a meta-analysis yourself if you want (like my own dual n-back meta-analysis on a widely-touted intervention which is supposed to increase IQ), you can interpret meta-analyses better, and you can draw on previous meta-analyses as priors (example: Jaeggi et al 2008 found n-back had an effect size on IQ of something like d=0.8. If one had seen one of the psychology-wide compilations of previous meta-analyses, one would know that replicated & verified effect sizes that large are pretty rare in every area of psychology, and so it was highly likely that their result was being overstated somehow, as indeed it turned out to have been overstated due to a use of passive control groups, and the current best estimate is closer to half that size or d=0.4).

I'm still reluctant to accept class grades and state-mandated graduation test scores as measuring primarily intelligence or even mastery of the material, rather than the specific skill of taking the test.

If IQ is the main cause of getting high class grades and passing cutoffs on tests and being able to learn test-taking skills (like learning any other skill), then couldn't the tests be measuring all of them simultaneously?