If many people used the same formula to convert their SAT score to an IQ score, I expected the line would jump out, but I don't see anything like that on the scatterplot.

IQs are often multiples of 5. I think that is the result of IQ tests that do not aim for precision and that conversion charts would end in any digit. 56% of survey IQs are multiples of 5. For those reporting both IQ and SAT, the number is 59%, so it is not depressed (or inflated) by those doing conversions. If we remove the multiples of 5, the correlation drops to .2 and stops being statistically significant. But both scatterplots look pretty similar.