Human beings are the only species that is capable of thought.
Homo sapiens is the only species capable of thought.
Human beings belong to the only species that is capable of thought.
'Human beings' aren't a species as such; they are members of a species, or even a genus. They are people.
Even "The human is the only species capable of thought" sounds odd, especially given that it's the genus Homo that determines hominids.
Was Homo erectus human?
The sentence in the original post indicates that human beings (are)(constitute)(belong to) a species........capable of thought.
That's what we have to work with. Homo erectus is irrelevant here; we are talking about present-day people. And we are focussed on English, not on evolutionary biology and its classification.
"people" can be referred to as 'who', but the operative word is "species", which is singular, and a "species" is not a "who".
That's why I would say 'Human beings are the only species (that is) capable of thought.'
Other thoughts are welcome. :-D