Having a child reveals sides of your partner you never knew existed. You’ll see what they’re like when they haven't slept, how they react when someone else’s bodily fluids are on their pants, and how they deal with a person crying for two straight hours (whether it's you or the baby). You'll also (or should also) witness them stepping up and supporting you in new ways; with diapers and snuggles and naps and in ways that push your partner outside of their previously lush comfort zone. For cisgender men who can't possibly experience breastfeeding, moms who nurse will see firsthand (hopefully) just what grown-ass men do when their partner breastfeeds.
Just because mom and baby are temporarily occupied, doesn’t mean dad gets to totally ditch all the other responsibilities that go along with parenthood. I mean, unless you guys have some kind of arrangement where dad can engage in those stereotypical "man activities" that our society has attempted to convince the masses only men get to experience and enjoy and indulge in (like video games and beer and endless hours of web-surfing) while you feed your child and parent solo (which, I mean, I'm not one to judge if you do have that set up in your home) then dad should be sharing the parenting role anyway that he can. Breastfeeding can be a good time for him to help and participate in and complete the other tasks of parenthood, as well as adulthood in general.