How old are both of you?
Just to summarise, you met at work and were good friends whilst he was with his ex? If so that's a red flag
You got together within 2 months, he lived with you because he after the ex he was living with his mum, who was likely to be providing childcare. Another red flag.
After 2 years you had a child and give up work, did everything at home and for his children but withdrew sex? Is that correct?
If so, 2 years is when honeymoon tends to end and reality bites. Many people, often men can't live in the non honeymoon phase and the grass looks greener. They switch off because they want the excitement that only a new relationship brings.
If he's staying could it be that he doesn't have a new "good friend" at work to move onward to?
My instinct (and others have shared) he's the type of man who has multiple relationships but can't love, so when the going gets tough he will find an escape. However if he can't find someone else to wash his socks and make dinner why would be move out?
You on the other hand are feeling overwhelmed by guilt and want the family unit to continue, which is very understandable. He will continue to play you until you get tired of it or he finds another "good friend", preferably one with a place to stay in.