I agree... it's Murphy's Law.
My mum used to joke that she only ever "believed" any of us "not feeling good" at 5pm on Fridays. Because that's when kids get sick for real. The moment the pediatrician's office closes.
Ped offices not being open on the weekends stems from an age when doctor's did housecalls. Office hours were M-F for a select period of time, and then after hours parents called their docs and the doc and/or his nurse would come to your home.
Fortunately, since that era ended over 40 years ago, more and more offices have weekend hours. (Actually, my grandfather still did housecalls until 1995, which is how I know the complaint about office hours and housecalls from the med perspective). Our own ped office the docs only work 3 or 4 days a week... but it's staffed 7 days a week and there is always one of them "on call" for after hours at the hospital. But a large part of the reason why more offices don't is that practices have to remain solvent. When they see 100+ people a day, but only 2 people at night, it doesn't make sense for them to be open at night. Same token, most docs want to work full time. It's almost impossible to ONLY work full time and cover 7 days a week. There's either an extra salary to pay or overtime to pay. And after 15 years of school, 7 of that working 24-72 hour shifts at the drop of a hat, and having to use IV's and stimulants and sleep in cleaning closets... most docs who have private practice out and out refuse to work more than 40 hours. "I've served my time." type of an attitude. Our office is an oddity (the docs all only want to work part time so they can be SAHPs with their children half the time), so they don't have to pay salaries to more docs for extra hours... they just rotate. One set works 3 days a week, the other works 4. Each gets a weekend day.