I’ve heard that in Spanish latinx is pronounced as if the x were an e

That's mistaken, there's latinx and latine and only the second has an obvious pronunciation.

They also mentioned that the -x is preferable to -@ because it includes non-binary.

I don't even know under what logic this could be true. What I have seen is people saying that these neologistical endings in general are preferable to the more traditional -a/o or -o/a, because these imply that you can be in a category who ends with an -o or an -a, whereas the other ones don't.

I think the reason -x is used instead of -e is because adverbs often end in -e.

I don't think people think in these terms. There's lots of nouns that end with -e.