Correct, cache exists for that reason. But youre still loosing time and space by saving it to memory ( cache is just faster acccess for the cpu. Its still in ram or in the pipeline to be pushed to ram on next flush ).

Also true, per thread you would need more memory to save the cpu's state when switching threads. Now i kinda want to do some calculations and tests to see at what point it gets better.
I always figured that per thread more memory is needed, but that the thread itself needed less memory ( or time to access it ) because it can contain more temp values in the cpu's registers.

Again though, there claim is bullshit or not totally depends on the kind of work youre doing and for video work i consider it bullshit as well :')