I know this started as a post-and-run, but I might as well add my own empathy for the OP, as I'm in the process of trying to get a renewal done as well.

In most cities, including our tiny little hamlet of the greater Kansas City area, the post office is the ONLY option. And since the USPS has no mechanism to book passport renewal appointments online, you must pick up the phone and physically call each PO on the list.

The problem is that of the 15 post offices I called, only 1 of the numbers actually functioned. The other 14 either had an old-school busy signal or just rang and rang and never kicked into any kind of voicemail or automated response. The one that *did* work was actually answered by a pleasant person who said they do 4 passports per day, and tomorrow's booked. She connected me to the consumer affairs person for the whole region - another pleasant enough person who is currently trying to find out if any post office in the entire city can do the renewal for me. (She apparently has other numbers to call than what's on USPS.com.)

There is one PO in the entire area that does not reference appointments at all, but I'm not sure I trust it since so little website information seems correct to begin with. It's in a far southeast suburb...I'd hate to drive out there to find out they do require appointments or don't do passports at all.

Worst case, we will all be in Los Angeles in 3 weeks, so I am fairly certain I can find a walk-in processing center there.

Our company has a vendor who can handle all sorts of business-travel passport emergencies, but renewing one for someone under 16 is something they can't handle. Apparently dealing with the post office can't be avoided.