the irony is that this kind of performance rarely pays off long term. I've had managers/engineers and senior managers/engineers pull this stunt thinking that looking overwhelmed will create a good impression but it doesn't and it often stops them making the next level who perceive them as useful bodies but not bodies to entrust with more complex decision making and relationships.
Its also a very handy tactic for avoiding the more tedious aspects of parenting a baby. What happens when you go back to work? Or is your career to be sacrificed on the altar of Mr BigCheese?
Absolutely this. People with such poor work habits, who are this performatively overwhelmed, aren't going to manage to surpass the 'useful drone' stage.
And absolutely, OP, you need to think about your return to work, and making it very clear that he's just as liable as you for the kind of stuff that happens when you have two careers and a small child in childcare -- covering sickness, emergencies, medical appointments, childminder/nursery closures etc. DH is CEO of a big organisation and he still manages this stuff. Because nothing exempts you from parenting.
I've said this or something similar on Mn before, probably under a different name, but a friend of mine was very like your husband, minus the eternal boasting -- he was never off his work email, he went in at weekends, he left the house before 7 am, and got back around 8 pm. He continually huffed and puffed about his workload, and gave the impression he was irreplaceable and running the place. His wife, who worked in an entirely different field, believed him implicitly.
It was only when I started meeting her that I saw the scales gradually fall from her eyes. Because I did the same job as her DH, working 9 to 5 three days a week in the office and two days WFH, working efficiently. I think she finally got it when I was promoted above her DH. I'm very fond of him as a human being, but in work terms, he's a faffer who wastes time, is disorganised and inefficient, and uses work as an alibi for not parenting and being involved in family life. He was considered a liability in our shared workplace.
He's now also divorced, and having to deal for the first time in his life with day to day parenting, cooking, cleaning, playdates, homework etc etc when the children are with him.