"running it through an LLM" doesn't mean "Give LLM my text -> Copy-paste the output of the LLM" does it?

The article seems to imply this is what is happening, as writing style converges towards LLM's style. You can call it what you want, but the important bit is that this is how it appears that LLM's are being used.

Checking against an LLM then using your own voice feels completely fine

Why use an LLM? If you're worried about style, starting with your own voice is more efficient. If you're worried about facts, looking something up in a primary source is best, and is probably cheaper on a few axes, especially if you need to check/validate anyway...