Here's a nice little list of things he can be doing as he cant do ONE thing (feeding):
Cooking (inc prepping snacks for you)
Tidying & cleaning
Laundry (always plenty of this in a home with a newborn!)
Grocery shopping
Other shopping as needed
Changing beds
Nappy changes
Soothing baby when not being fed
Keeping on top of household admin inc finances
Washing/sterilising dummies if you're using them.
Bathing baby
He needs a kick up the arse!
When I was 3 weeks pp we'd both only been out of hospital under 2 weeks, then dh was back at work full time (army, not some cushy office job, also meant we weren't near either family), but he was (thankfully) working roughly regular hours. So as he was an early riser, he kept dd occupied as he got up and ready for work allowing me a little more rest in mornings, then handed her over to me when he left for work. I did what I did during day, when he got in from work I handed her to him he played/cuddled with her, then did her bath and "bedtime" routine while I tidied and made dinner (rarely major recipes from scratch at this point, simple dishes and probably too much beige/freezer to oven stuff) after dinner she fed and he'd do dishes, clean kitchen and put a laundry on (I think we were easily doing at least 3 a day at this point), baby would go down to sleep for a few hours and I'd go bed early, when she woke around 1030/11pm I'd feed then he'd take her so I could get more sleep/rest and when she fell asleep again around midnight he'd go to sleep. When she woke in the night (we all co-slept in king size bed. Seemingly verboten now worked for us!) I'd feed her but if she needed a nappy change or was fussy he dealt with that (when she was colicky she seemed to prefer a weird position on his chest with him sat in armchair).
Weekends - we took turns having lie ins - he had sat as a lie in, I had Sunday's. Though of course if she needed fed he'd bring her up.
YOUR baby's dad can & should DEFINITELY be doing more than he currently is!
Him working outside the home doesn't mean you're sat on your arse twiddling your thumbs!
"You ignore the grumpinesss. It's training for ignoring your toddlers tantrums." So true! Which is also training for ignoring teen tantrums!
Gym?!! No! He needs to be at home fulfilling his responsibilities as a parent and partner right now! Wtf!