I don't know how they pull it off. I agree with you - my wife also stayed home until our last kid went to preschool. Then it was off to part-time work (teaching) for her. She lost 20lbs within 3 months from the reduced stress, and was in a way better mood. Being a stay at home parent is pretty friggin tough, but on the other hand, day care for multiple kids in CO... you could be looking at $3-5k/month. You'd probably need a job that pays $70k otherwise the math works out better to stay at home, so I don't blame his wife for staying home.
Financially they must either be drowning, or getting money from other sources. Even if Seth pulls in $100k/year from his running "career" (I would be shocked if he made this much)) literally anyone with 6th grade math abilities knows that doesn't pay for a $900k house and 6 dependents in very HCOL area.
With regards to your last point, I always made sure that my wife had at least 1 hour alone every weekday and 2 hours a day on the weekends. Even if that meant me skipping a run or staying up late to finish other work. She didn't always take me up on the offer, but would sometimes go for a walk, go to get her hair done, visit a friend, walk on the treadmill, mess around on her laptop in our bedroom etc. She later told me it was often that short break that kept her from going into a total depression.