I don't think that people necessarily 'soldier on' out of some desire to prove anything to colleagues but perhaps are aware of the pressure that absence might put on them
^^ This. People saying he should take as much sick leave as he feels necessary clearly haven't had to deal with the fallout when you work with a shirker who takes a day off sick at the drop of a hat, having to do their work for them as well as your own. Soul destroying and you lose sympathy fast when you're repeatedly in that position.
While it doesn't seem like this man is a shirker, there's a lot he should be doing before calling in sick. Using his annual leave, not everyone needs to book leave months in advance and even if he normally does, if he explains the situation, his employer will probably be flexible.
Can he work from home so he can be flexible about when he works and doesn't need to drive?
Are both parents sharing the overnight load and catching up on sleep where possible, using earplugs if necessary. The parent who is awake could take the baby out for a couple of hours.
If I was his employer, I'd expect all that to be done before taking sick leave due to tiredness.