Depends on which naming convention you subscribe to. Most people just use H. neanderthalensis for convenience rather than writing out H. s. n. since everyone knows what they mean either way. Neither classification is obviously correct either, hence the introduction of more nuanced categories like "allotaxa" that aren't reflected in binomial nomenclature (yet?).
I think the contextual clues are enough to correctly interpret the current wording.
For the informed reader. But it also implies that perhaps Neanderthals were not human, which is completely false.
Imagine being triggered by people
assuming someone’s species.
Not sure what is your angle here, but Neanderthal are considered to be "archaic humans" and while it is not technically wrong to write "humans" instead of Homo sapiens then it would be more correct in this context to be more specific and write "modern humans" in my personal opinion.