My career hasn't gone the way I planned it to and there's been lots of times I've felt like giving it up just to be with the kids. For most of my twins first three years, I've been gone, either doing on-call charter or regional pilot. Neither pays well enough that we couldn't cut back and be on one income.
But I don't think I'd be entirely happy being a stay at home Dad either. But, they are three now and are just so big. I feel I missed a lot being gone as a charter pilot and now as a regional pilot. It's time that you can't get back and it weighs more on me each time I go out, especially on 5 or 6 days. I try to spend as much time with them as I can when I'm home. As dhood mentioned, Skype and Facetime help, but you still don't get to hold them every night. I knew all this would be part of the deal, but until you are actually doing it, you don't realize how much being away can suck.
Having a supportive spouse is a big deal. Supportive not only emotionally, but financially too and I don't mean that lightly. I work with many people who make less than their spouse, which at the regional level, isn't hard to do. Money is a big factor and kids take a lot of it. We could make it on one income, but it would be tight and we don't even have many expenses and little debt besides the house and one car.
Kids are never easy in this job. As an instructor, paid by the hour, I worked my butt off during the 6 months of the year you can fly nearly all the time in Pittsburgh. That meant leaving at 7am or earlier and not getting home until midnight sometimes, but at least until 8pm many days. The money stinks unless you're at a really busy school or paid on salary. Now I'm home less, the money is better, but still not seeing my kids most nights except for Facetime.
This is rambling a bit more than I want, just because it's not a simple issue. When you're a pilot, your job is to fly people and/or things places. That means you will be away from home. Working a CFI job is more like an office job, but I never found the risk/reward of teaching people to fly high enough to consider it full time work, it was always a stepping stone job. It's all in what you value more, a flying career or kids and family. If you want both, you and your family have to make sacrifices and it isn't fun.