The claim is predicated on the mutual understanding that being a white nationalist is bad. If you lack that mutual understanding, there are bigger problems at hand!
There is no way to be reactionary without being a white nationalist and the two are interchangeable but you chose one instead of the other for some reason?
Sure. You can substitute “white nationalist” for “any kind of reactionary, including white nationalist” in my previous comment, if you find that more palatable.
There are certainly places in the world where there are significant local reactionary ideologies that are not White Nationalist, but the US is not among them.
> Why is it bad if it pulls people to the reactionary?
Because, and to the extent, that the thing it pulls people toward is bad.
> We might as well say the glamorization of cosmopolitanism is a recruiting tool for neoliberalism.
That's less true, though, but, yes, glamorization of cosmopolitanism feeds a bunch of ideologies that are not the same ones the “trad... ” movement feeds.