Everyone is a shitty parent in some way. All of us make mistakes and are selfish at times. It has got to be hard when it is not your kid. I can't even imagine how hard it would be when you don't have that love there. If you really want to make it you just have to keep giving. That dynamic you're seeing is a natural dynamic of a family. The mom wants to baby him, the father steps in and helps the kiddo learn how to trust and do things on his own (from someone that is not mom, your role is important). By someone that is "not" mom and more importantly a different gender, that relationship helps him learn how to trust others (outside of mom) and himself. There are a lot of kids that never get to learn that lesson. If you teach him that from a place of love and compassion (and most importantly integrity) he'll learn that stuff. It takes time. If the kid sees that you are cold, and stern he will see you as a hypocrite and a fraud and later see the everything as a fraud. Kids just know what love is and what it is not, bino. I can not stress how important the "not mom" role is on a kiddo. If you're a fraud, he can see it better than any adult ever could. If you don't interact with him with compassion and integrity from your heart, just forget it. It's too important to take lightly.
This is just how a family with children works. Let your gf know that it's natural for you to want to not "baby" him.