Nerd Version:
On the one hand, Geese has been portrayed as an evil bastard consumed by his own ambition for power.
On the other hand, it's the only reasonable explanation for why he doesn't keep Rock closer to him in any of the games. Well, that or he never fucking cared. It's one of those two.
Real Bout's 'canonical' ending humanized Geese, in a sense, by making his rivalry with Terry personal rather than Terry and Andy just being two more victims on his way to achieving dominance. Rather than allow himself to be saved by Terry, he chose to hurt Terry by refusing the aid and plummeting to his own death. So given that, it's possible that Geese has more feelings than simply the desire to attain power, and he's more complex than any of us ever really thought about.
EDIT: Since Geese had a relationship with someone, and that someone got pregnant, it's entirely possible that he chose to bury those parts of him because they would have gotten in the way of his ambition. Having a son around would have given him a new responsibility outside of the scope of his goals.
Normal Human Being Version:
Martial arts anime stories are cheesy and weird. Don't think about it too much. It's not that deep and theorycrafting these games in 2026 is gay.