I do think the issue of Original Sin is a matter of faith and morals, and that was his primary concern with this encylclical.
I agree entirely, so I must not have been very clear in my earlier comment. The statement from HG:
the faithful cannot embrace that opinion which maintains that either after Adam there existed on this earth true men who did not take their origin through natural generation from him as from the first parent of all, or that Adam represents a certain number of first parents.
As a statement about
true men, it is a theological statement about he relationship of God, original sin, faith, etc.
There were other hominids, members of the same or related species, who lived after Adam and did not take their origin through natural generation from him. There were multiple hominid parents who contributed to the development of hominids. This is a scientific statement.
We are inclined to read HG as affirming the theological statement, and condemning the scientific statement. But Pius XII had no particular competence to declare scientific facts are true or false. We have to respect the theological opinion, but we do not have to accept his rejection of the scientific statement. If we read HG as a rejection of the scientific statement, it would be a statement about science, not about faith or morals, it would not be an infallible doctrine as a scientific statement, even if the theological statement is.
Sorting this is difficult. As Pius himself says, it is difficult to see how both the theological and scientific statements above can be true. That is why we are discussing it now.
On another issue, I would count the rational soul as exterior to the genome, though the genome has adapted to it. Any genetic changes are after the soul, by natural selection to improve our use of our rationality.