As the father of three boys (who can just flat wear me out) I can totally sympathize. Children can be exhausting, and combining that with the compulsion we all feel as photographers to just shootshootshoot all the time, you are left feeling, well, wounded, as you said. We all want to be out there in the world living through the camera. Anything that prevents us from doing so, we resent.
Honestly, you have a lot on your plate now, and stressing over not being able to shoot is not going to help you or your wife or your kids. You are probably going to have several down years of photography.
Robert Adams once mentioned in an interview that he hadn't photographed in several years due to work on his books. But he was still a photographer, obviously. You can either hit the pause button, or you can hit the quit button. You're in a space where you will need to hit the pause button or certainly slow down and find what you need to photograph much closer to home. In the last two decades of his life, Paul Strand made the most incredible flower and garden studies in his own backyard. Maybe it's time to start working on still lifes!
The other thing I would suggest is to get those photography and art books off the shelf, read them deeply, and learn more about the history of photography. You can work that way and still be educating your eye and your mind, so when the time becomes available, you are ready to move forward quickly. Sometimes not shooting can be clarifying if it gives you time to contemplate new ideas. We can all compulsively burn through some film; sometimes we don't spend enough time thinking about why we do it.
Being that my boys are now 12, 9, and 5, I am in that later phase where I can carve out blocks of time to work on the photography. Just this week, several really fortuitous circumstances have aligned on a project that I've had in mind for several years. It's about to explode out of my head and onto the paper, so to speak.
So, don't sweat it, just enjoy the time with the kids because you'll turn around in ten years and say "holy crap, where did all the time go?"