A lot of this is the difference between having you first child, versus subsequent kids. With my first one, everyone's anxiety was high. There's a level of uncertainty where you don't know exactly what's coming or what it's going to be like (and you think you know, but you don't). But you do know it's going to be a huge moment in everyone's life. Because of that, the temptation is to clear the deck, take a bunch of time off, to help out and enjoy the moment.
With my second kid, I was doing what
@Falconclaw is doing, and trying to prepare ahead of time, but we were infinitely more prepared. I remember a colleague of mine, who by then already had four kids, saying, "Why are you going to take time off for a baby? Are you going to breastfeed the baby yourself? Are you going to stare at it for the 22 hours per day it sleeps, are you going to stay up all night burping it, then work all day the next day with no sleep, for weeks on end? After being off a week, then what? What changes? Doesn't the baby still need all that? Don't take time off. Save the week off for later, when you can use it."
It seemed like an outrageous attitude at the time and I don't agree with it.
Then baby #2 comes. I had a full week off. I was in turbo-dad mode. I offered to do night feedings with pumped milk that week, and nighttime diaper changes so my wife could get caught up on rest after labor, and all that. My wife's response was, "Don't bother. I have to be up every 3 hours to either feed this baby or pump, anyways. I might as well slap a new diaper on, too, while you sleep. There's no point in both of us being up all night, and zombies during the day. You sleep at night. Your job is to do as much as possible during the day time, in case I need to nap from the total exhaustion of being up all night."
Then, during that whole week I did as much as I could, but my wife had the whole "new baby thing" on such lockdown, there wasn't as much for me to do as either of us had expected. Mainly, I hung out with our then 2-year-old, keeping her out of trouble and did some chores, food, etc, and watched the baby if and when my wife napped. Then at the end of the week, it was off to work, she stayed home with the kid, and before you know it, it's all flown by and your kid's addicted to youtube.
At the end of the day, what you want to do, is whatever makes your wife most comfortable and whatever will allow you to look back later, and say you did the right thing. And what that is, might change with subsequent kids, if you have them.