I don’t think that the comparisons i.e. “well there are a lot of men who piss off and never see/support their kids so it can’t be said that women hold all the cards,” are helpful. Two wrongs don’t make a right and all that.
I think it certainly is true that there is an assumption that when a couple splits the woman will be the primary carer and the man will get to see his children on her say-so and for anything else he should go to court, and there are certainly women who do use their children as pawns against their ex’s or who restrict access on the basis of how hard it is to not see their children.
When me and my ex split I was certainly told by some people to “put your foot down” and “tell him he can’t have him for that long,” when e.g. xh wanted to take ds away for a week to see his parents over new year. 
In those instances I wholeheartedly agree that it shouldn’t be the woman who automatically gets to make the decisions and that essentially where possible, if you have a child together, you parent a child together.
But that kind of scenario isn’t what the op is talking about here, is it?
Op says that her dp has his children two nights during the week and every other weekend. Op you do know that in the eyes of the courts that is considered a 50/50 arrangement, don’t you? So what more is it exactly that he wants?
You say that his ex isn’t working, presumably she is the one who does school pick-ups/drop-offs/i the one who is home with the children when they’re ill? Yes, she could get a job, but if that were to be the case, would your dp take the time off at the drop of a hat to look after his children if they woke up ill at his in the morning?
If he’s not in a position to reduce his hours (and tbh i don’t think that should be criticised) there is no merit in having the children on a “50/50” basis just so that he can say he has them 50/50 while actually paying someone else to look after them. That is point-scoring, nothing more, and doesn’t show him in a good light at all.
As for the rest, it could perhaps be argued that women might want to consider going back to work in order to be financially independent. But you know, bringing up children is just as valid, and not all women want to go back to work and leave their children in childcare, and if financially the couple is able to afford for the woman to do that, then why not? The only thing would say is that women do need to be aware that if they give up a career to bring up children, they will likely struggle to get back into work when they are ready to do so, but that still comes down to personal choice. I couldn’t bear to put my baby in nursery when he was tiny, and financially we could afford for me to be home. Plus because xh was commuting and I don’t drive, all the childcare fell to me and nurseries/public transport didn’t tie in with each other when I did consider the possibility of going back to work. It’s just not always that black and white.
As for not moving a new partner in. One of the reasons why couples split up is because they are unhappy in their current relationship. Any couple has the right to move on and find that happiness with someone else. Do I like the idea of another woman living with my ds when he’s at xh’s? No, but equally I believe that ds will benefit from having relationships with multiple people in his life, and as long as she is good to him who am I to say that xh can’t find happiness with someone else because of how I might feel about it? Similarly I know that xh finds it hard seeing ds with my dp, but equally I have the right to move on and have a relationship in my own right. We don’t actually live together atm because of geographical distance, but the only person I would make a decision for would be my ds, not my xh, as I’m sure my xh wouldn’t be considering me when moving in with his dp.
Don’t think your ex will appreciate you martyring yourself for his benefit, in fact he’ll probably enjoy the fact he still holds that kind of power over you.