You say you are not legally a step parent because you are not married, but it doesn't matter. You all live together. You invited him to move in, you had a child with him knowing he already had a child and knowing he wasn't a high earner. What if the boy's mother had died suddenly? You'd have two of them full time and really wouldn't have grounds to complain, would you?
I get that there are bigger issues over how much he contributes financially and in terms of practical help in the house, but that's a separate issue that needs addressing.
Some people feel he's played you. Moved in to a nicely feathered nest that has made everything easy for him, provided his son with a comfortable home life that he could never have provided by himself. Some people seem to think that make him a cocklodger, and for the lack of help around the house alone, perhaps it does. But is he doing anything any differently than hundreds of thousands of single mums do? They find themselves a nice 'responsible' hard working, comfortably off man (who is nothing like their feckless ex, which is the whole point) to take care of them and their existing children. Then they make sure they have a baby with him so that their feet are well and truly under the table. There are many, many men taking full responsibility for children that are not theirs because the bio dad is useless,poor, feckless and/or absent. Is it the end of the world if it happens the other way around sometimes? You knew the score when you took him on.
If you aren't happy and you'd rather live alone with your own child or children, then fine. Do it. But if you want him then you have to accept he comes as a package deal. You don't get to sideline a child, when they are happy and settled living full time with that parent and that parent is happy to have them there.
If a man moved his female partner in, had a baby with her, then told her that he wanted her older child to start going 'home' for longer periods of time to his dad's to give 'their family' more time without him, knowing full well that the dad was flaky and largely uninterested in having him there, there would be uproar.