That's pretty much exactly the claims that the strongest proponents of IQ tests are making.

I lack the skills to determine that, so I am limited to trusting the people who tell me the tests are "pretty good but not perfect".

Source? Who is claiming the having a high IQ makes you better at for example athletics?

To pre-empt you expanding that to things like running speed, I considered it too obvious to bother mentioning that I'm taking about mental tasks.

For that specific example, athletics, there does appear to be a correlation. It may be any or all of A->B, B->A, and C->(A, B), but "healthy body, healthy mind" is a meme that appears to have a similar age as the concept of writing.

Finally, as I have been specifying IQ 85 as the "huh interesting" threshold rather than "high" IQ, I'll ignore your goalpost shift and say that your citation is me right now saying: I reckon IQ improvements in at least this zone (say +/- 0.5σ) is causal for better outcomes with regard to [avoiding injury, pushing performance limits, maintaining training regimes, learning techniques], all of which improve overall performance.