Your "extremely likely" scenario is the "cold winter theory", which is problematic for many reasons including the myriad of historical instances where hot-climate civilizations outpaced cold-climate civilizations.
How would you explain how geographically distant groups in radically different conditions could, over millennia, converge on all of the exact same non-visible traits without even minor variations? I truly can't imagine a way that this would be possible aside from the infinitesimally unlikely product of completely random chance.