He also dreamed of having a family with Casca right before the eclipse situation. That’s what he desired in his heart of hearts.
I don’t think there’s romantic/sexual conotations to Griffith and Gut’s relationship. Miura wouldn’t have hesitated to bring that out more. I think Griffith wanted normalcy (like that family life depiction), but can only see people as objects the way people have seen him, and it’s impossible to achieve that with that outlook.
So he attempted to take it by supernatural and incredibly evil means.