So it's in the developer's interest as well for the manager to figure out a way to reshape his/her job to make it sustainable (and thus keep the paychecks coming) rather than just saying "uh, sure, go deal with your problems, take as long as you need; when you run out of money we'll still have your job here for you."
Plus -- "first newborn in the house" is something that causes upheaval for maybe 6 months or so, and then life settles down again. If you have a supportive/flexible work environment then "up all night with a crying baby" here and there (before the baby grows out of that stage!) doesn't result in crushing stress and visions of disaster.
Edit for more context -- we have a developer who works part time and sometimes takes long sabbaticals -- switching into his alter-ego as a theater director. It's quite a different sort of situation from someone who has a newborn and is burning out.
Ha! That's fine until they get to 8 months and stop sleeping through the night again (which apparently is a common thing). A few weeks of chronic sleep deprivation can absolutely destroy your ability to do anything vaguely brain-work-related.
I have two kids, 3 and 6. The second was hard, but we knew what were getting into, largely, and how to survive it. The first was... rough.
I'm not sure about sleeping through the night stuff -- we did the co-sleeping thing, which helps a lot! But still, by 8 months I'll bet the parents would have some strategy in place -- maybe "hey, tonight's your night to wear earplugs" coupled with strategic naps.
If you're not going to do it, who will?
The work might wait a week or two (vacation), but do you really think the clients will be OK waiting months (sabbatical)?
The only industry where it's acceptable to just take off is within academia and that's mostly because most of the teaching is done by staff that would never have the option of a sabbatical to begin with.
Time off is not a solution. The problem will simply resurface when the person comes back.
The solution is to re-organize the work. Hire an additional person, if needed. Add work that the employee will enjoy and that will be relaxing. That's a solution.
If the company isn't interested in solutions, then the solution for the employee is still not a sabbatical. The solution is to quit and find a company that isn't so myopic.
And if you're hit by a bus, then what? The company goes under? People shouldn't exactly be replaceable, but they also shouldn't be carrying the entire company on their shoulders. No wonder they get burnt out.