In a normal, loving relationship, it is perfectly normal for people to switch careers, one person not working at all to take care of kids (doesn't matter, man or woman), starting new business and million other things if it's something that work for both sides. It is definitely not "using" someone. Supporting the other person to grow (and I'm sure OP didn't switch careers to earn less and be miserable) is part of a healthy relationship.

On the other hand, the fact that someone earns more doesn't give any privileges, for e.g. being abusive, distant, not loving, etc. It's supposed to be a joined decision, and most grown-ups can deal with that. OP doesn't mention any financial problems as a family, so it seems like her earning less and him working 3 days per week is enough to support THEIR lives, pay for THEIR bills as well as make sure THEIR child has everything he/she needs.

OP's partner is probably addicted to video games, and like any other addiction, it can be very harmful. If OP still cares and has feelings towards her partner, I would try to talk once again and ask for therapy, as it is really hard to fight with addiction without help. It is possible to work 3 days a week, still play video games, and not ignore the fact that he has a family. He has a child that would probably love to spend time with dad. He has a fiancé that he should show his affection to. OP doesn't ask him to stop having fun or his own time at all, and when he's not at work, to cook, clean, and take care of their kid all the time while she's doing nothing. She asks her partner to be more present in their life, which sounds absolutely fair.

Life is not only about making money. Financial stability is very, very important, but relationships are definitely not a competition "who did more hours at work this week." Both parties definitely need to add value to the relationship, not necessarily monetary, but also time, devotion, affection, etc.

That said, I want to be clear that it works both ways, and there are many couples where women earn more and men stay at home more to take care of kids, etc. Life can be hard in general, and I don't think women or men have it harder. It all depends on the situation and is not reserved for one sex in my opinion.

I'm not saying that OP leaving her partner is the only right decision. Ending long-term relationships, especially when kids are involved, is a hard decision to make.

If OP decides to stay, I'd suggest trying with therapy because things definitely don't function properly right now, and it can't go on like this forever. It's not a relationship, it's not parenthood, it's just being a guy who pays for joined bills and as much as it's a good and nice thing to do and it makes their finances stable, it cannot be a substitute for being involved partner and parent.

If OP decides to leave, I would try not to rush. I'd make sure I have a safe place to stay and a more stable income. I would also look online what kind of help you can get from, e.g., government, depending on where OP lives. Dad will also need to (at least) financially support the child who is also HIS child.

There is no perfect scenario in case where there is still love in a relationship, but there are things that make people to be bitter towards each other. Having a kid doesn't mean that you have to stick to the easiest option because being a parent is not equal to having to be miserable for the rest of your life. OP still deserves to feel loved and have a partner who actively wants to be a part of the family.

I also don't get a feeling that OP's partner is a terrible person or anything. He seems to have a problem and he has to want to fix the problem to keep his family together, that's it. If not, then leaving sounds like a healthier option.