I can't speak for anyone else, but for my family, it's much clearer for everybody this way. When I'm at the office, I'm working. When I'm at home, I'm home. My wife doesn't mind my working long hours at the office, but when I'm home, she and the children get my complete attention.
I've 3 pre-K kids and have worked at home since before all of them arrived. The kids know that when my office door is closed - I'm at work.
Overall - I find working at home with family to be less interruptive and distracting than an outside office.
For the initial recovery period I found that being in the same room or nearby out of a desire to be helpful was a bad idea. (I don't want to seem like a jerk here, but I have things that I absolutely have to get done. No paternity leave for me.) Simply because my wife knew I was there, she would ask me to do more and more small things for her, i.e. walk across the room and get that blanket for me. Frequent little distractions are a productivity killer. If I was downstairs with a baby monitor, the requests would be a little less frequent. The optimal solution (if I wanted to actually get anything done) was to work outside the home but within about 10 minutes. This way, if she really needs something, I'm a phone call and a short drive away, but generally she has to remember to ask for things she needs when I'm home and non-essential tasks can wait.
So, I'd recommend finding a place close by where you can work on your own, and take breaks every couple hours to come home and help with anything.
Now I have a permanent desk in a coworking space that's five minutes from home, and it's better for everyone.
I now rent a desk out at an incubator and it's been much nicer. I have a clean separation from work and home and get showered and dressed each and every morning like a normal person! :)
Even though I'm a FT employee right now, I'd probably still try and go this route even if I was freelancing.
This was critical at the time, as neither of us was in a position to provide primary care to the babies during the day. We were really happy with this and our kids have done well with daycare.
When my wife is home I find the "door closed == serious work time" method and noise cancelling headphones do the trick. The most important thing is that your partner/children know and understand that your work space is a place where you work. My wife knows that I wouldn't be able to do X if I was in an office elsewhere, so I probably can't do X either while I am working at home.
On a side note: A very nice benefit of working home w/ family is the ability to take breathers whenever you so chose -- completely remove yourself from your work environment for a few minutes and (hopefully) change your perspective/attitude to a more beneficial one.
Second, I head out to a co-working space or cafe a few days a week. Getting out of the house for the change of scenery is a huge energy boost and it helps to interact with other working people.
A couple of notes of taking care of the baby while working: (1) The newborns are on short cycles but if they're mild babies, you actually do end up with lots of time on your hands. You just need to be focused and get to your task right away since you'll likely be interrupted in an hour or so. Break things up into 30min-1hour chunks. (2) I tried taking care of our 6 months old full time while working at the same time -- doesn't work. I got maybe an hour of work done a day. If this is your scenario, hire a nanny!
Once a routine is formed it is still very hard, especially when you little boy finds his way into your office with a storybook in his hand wanting you to read it to him. Expect to be doing some late nights when the house is finally quiet.
Perhaps you could consider investing your first considerable revenue in an outdoor office so that you're actually "leaving the house" when you go to work. That's what I'll be doing...
Also my wife found it much more empowering to just have to deal with the baby on her own instead of having me as a 'last resort' option just a knock away.
I then asked why he didn't consider this possible impact on his livelihood( the productivity aspect )before deciding to have a kid.
So, to use your response as a template, I would rewrite with the following:
Let's say in 10 years he fails to get his company off the ground because 4+ hours per day that would have been previously available to him to work on his project were not, due to the presence of a child. The kid has to take out student loans, parents who have nothing to show for 10 years of stress - and if the OP had to, he could take a second job and spend what would be kid-hours with his work.
Each to their own I guess.