In general I can sympathize with the frustration of trying to parent when your child has to go to another parent who won't work together for the majority of the time. I can't think it would be easy to teach a child to sleep on their own if more than 50% of the time they are with a parent who says they don't have to do it.
All that is going to happen is they are going to kick off, and then that puts you in a very uncomfortable position of having to be the bad guy, in a situation where they can choose the EX as the good guy. Its a very common blended family dilemma...
That being said, I agree with the sentiment on here that your partner doesn't look good in all this. It does feel like he wants to excuse is lack of effort by blaming his ex. It could well be true that his Ex is a lazy/bad mother, but if that is the case, a good parent would fight for more time/custody, put more effort in.
Sitting back and watching your kid fail and blaming an ex isn't what a good parent does.
Overall though, poor kid. If we take your word as truth - he has a mother who isn't very competent, a dad who knows better but can't be bothered to fight for him, and a dad's girlfriend who might be capable of raising him right, but isn't in a position to do so, and as such is just annoyed at the whole situation. That isn't an optimal environment for a kid to grow up in.