Speaking from experience: try this for a few months before you commit to it! I worked from home for 2 years, but 6 months after my son was born I chose to switch to a job with an office. It was impossible to get anything done at home with the constant interruptions (even though I wasn't the primary caregiver). My work and family lives are both loads better now that I have proper separation between the two.
I have the opposite experience: I have 2 kids (one 2 years old, and another 3 months old) and I work from home and remote 100% of the time. I _still_ get more done at home than I ever did in an office, even with 2 little ones who are taken care of by my wife when I'm working.
Long story short, I first worked flex part-time (handing off time with my then spouse) when my kid was very young, then switched to being in an office when my kid started school and I was a single parent.
I went remote when my kid was in middle/high school because my kid needed me and it was good for me to be around. You can tell a 14 year old to shut up if you're on a conference call...most of the time. Also, I could take a few minutes in morning to get the kid to school (we lived close by to the school) or take a late lunch so my kid could go to after-school practices, classes, etc.
When a kid is in elementary school, it's vastly easier to work in an office. Elementary school is when life will be the most predictable with a kid. When a kid is in middle/high school, that's when you need to be present. For all the reasons you're thinking of. Teenagers are jerks.
All in all, I did pretty well. Kid is at a highly selective college these days.
Thanks for sharing your experience. I'm about ten years away from having kids (if I had full control over my future!), so I have a lot to learn. I'd just rather be close to my children than a long subway/car ride away.