So my hospital was traditional practice: clinic docs did Q3 inpatient call. It was also military: people put up with it because there was not an alternative, its just not something people do voluntarily. There are lots of people willing to work hospitalist/ER with nights and weekends, and lost of people willing to work clinic for 20 days out of 28, but its either or. Very few people want to work 20/28 days plus nights and weekends for the long term
There were rural civilian hospitals both in the area where I worked in the military and near my current civilian job, and none of them can find civilian pediatricians to replace the old guys who made the traditional model work. Some are transitioning to hospitalist groups, others to pseudo hospitalist arrangements (4 docs in a group each doing 1 week dedicated hospitalist and 3 weeks clinic), and some just keep raising the pay for their old outpatient guys to get them to hold on a little longer. Whatever they're doing it amounts to the same thing: paying more for more coverage. You are right it is a dead loss from a financial perspective to hire pediatric hospitalists for these little hospitals. There is a reason that critical access hospitals usually require state grants to stay open. They need to do a lot of things that operate at a loss and this is one of them.
I am curious what hospital could be described as both rural and desirable. Rich person ski country? Park City Utah, or similar? If that's true you might be correct about the group attached to your hospital, but your hospital is a unicorn. Most rural hospitals are fighting tooth and nail to keep any doc who is willing to stay, in any specialty.
Finally we haven't surrendered this job to the mid-levels. Lots of us do NRP call and inpatient. We just don't do clinic at the same. Just like an ER doctors and adult hospitalists I cover nights, weekends, holidays, admissions, calls, and codes. And then I hand the phone off to someone else and get time off. There are lots of pediatric hospitalists out there, if you want to hire them.