I think you'll indeed find, if you were to seek out the relevant literature, that those claims are more or less true, or at least, are the currently best-supported interpretation available. So I don't think they're assumptions so much as simply current state of the science on the matter, and therefore widely accepted among those who for whatever reason have looked into it (or, more likely, inherited the information from someone they trust who has read up on it).

Interestingly, I think we're increasingly learning that although most aspects of human intelligence seem to correlate with each other (thus the "singular factor" interpretation), the grab-bag of skills this corresponds to are maybe a bit arbitrary when compared to AI. What evolution decided to optimise the hell out of in human intelligence is specific to us, and not at all the same set of skills as you get out of cranking up the number of parameters in an LLM.

Thus LLMs continuing to make atrocious mistakes of certain kinds, despite outshining humans at other tasks.

Nonetheless I do think it's correct to say that the rationalists think intelligence is a real measurable thing, and that although in humans it might be a set of skills that correlate and maybe in AIs it's a different set of skills that correlate (such that outperforming humans in IQ tests is impressive but not definitive), that therefore AI progress can be measured and it is meaningful to say "AI is smarter than humans" at some point. And that AI with better-than-human intelligence could solve a lot of problems, if of course it doesn't kill us all.