Your opening post is fascinating. You value his feelings far far more highly than your own. He's top dog. The boss. Even though I bet you both think you are the one in control.
It is OK for a man to be grumpy you know. You can let them be grumpy. Man with sad face isn't a reason to panic yourself and scrabble to do whatever will stop the sad face.
See, he talked the talk when you were pregnant. Then he realised he didn't like the reality of parenting and the uncomfortable choices normal people make when they become parents. You then made a schoolboy error. You treated his reaction as acceptable and took responsibility for fixing his discomfort. Your posts are full of your solutions for his situation, his moaning. He's a grown man FFS. Stop treating him like your idiot child.
Forget deciding which job he should get. Forget finding ways for him to stop moaning.
You know what you want. You want what was agreed. That is him doing more days of childcare and more evenings. Demand that. He will moan. Do not engage "you'll have to work something out".
He didn't engage with your complaints did he? He didn't take responsibility for your feelings did he? He prioritised his and you allowed it. Encouraged it in effect. His sad face is an exemption from adulthood. Your sad face is not worthy of acknowledgement.
He has left it to you to work something out for you, and what you chose was stupidly long hours, most of the childcare and paying his share of costs for him.
Stop prioritising his feelings. Allow him to feel shit. Allow him to feel the intense pressure to earn money. Allow him to suffer the natural consequences of being an adult and to find his own way to resolve the issues. Currently he resolves it by getting you to pay the bills and do most of the childcare. And how? By finding a woman that will pick up his slack when he has a whinge.
You'll never be able to lean on him so long as you are carrying him around on your back.
Your problem isn't him moaning about how uncomfortable the piggy back is. It's the piggy back. Drop him onto his own feet whether he likes it or not.