I really don’t think the word spiritual precludes atheists, you could even read it as “the human spirit”. The point is just that you need some framework to ground yourself. Plenty of people have “spiritual” beliefs based purely around the many-worlds interpretation of quantum mechanics or even simulation theory.

I'm an absolute atheist and philosophical nihilist and I have done psychedelics and I've never had a problem with them. Typically find the experiences positive. So we have at least one counter-example.

Equating a belief in gods with spiritual discovery is also a take.