When you say "country school sites", I assume you're talking about 1-room school house type affairs (and not the size/look of the urban schools we think of today when we think "school") right? Ala.: "Norman Rockwell" paintings type of 1-room school houses, eh? If so, you've got the ending dates about right: Starting in the 1940s (& maybe into the early 1950s) those all went by the way-side, as the era of bussing came into play. Prior to the 1940s, if it was a 10+ mile ride to the nearest town or school, these 1 room schools were often put up, if 20+ or so kids from the surrounding farms meritted it. But nowadays, bussing takes care of all that, and country kids are bussed to school, ending the need for 99% of 1-room school houses.
I can think of a particular back-country road that stretches for about 100 miles here in CA, before it gets to the next town of any size. And studying the history, one discovers there was literally a 1-room school house every 10 miles or so. And the last of them was discontinued in the early 1950s at the latest, being consolidated into larger multi-room modern schools every 40 or 50 miles or so (or "in town", etc..)
My experience with hitting these one-room school house sites is this: Let's say the school operated from the turn-of-the-century, till 1950. Invariably, if you find coins, they always seem to be on the latter end of the date spectrum (ie.: 1940s/50s). Rarely did we find coins from the earlier range of the school's existence. At first, we thought maybe it was just the luck of the draw, or a depth thing, or someone got there before us, etc... But soon we realized that some of them were quite virgin (d/t the easy 4 star signals like suspender buckles, harmonica reeds, bullet shells, buttons, etc...). So where are the OLDER coins we thought?
The answer is this: It wasn't till after WWII that American prosperity was such that little kids (esp. country kids, maybe not as applicable to urban-city kids) took money to school. Prior to the school lunch programs that started in the 1940s/50s, what need was there to take coins to school? Sometimes you hear an elderly person who grew up in the '20s or '30s joke that they never had a nickel to their name, or didn't take money to school, etc... I used to chuckle at that, thinking, like most non-md'rs, they just didn't understand, or were spinning tall tales. But the finds in the field do tend to bear that out. I've even seen this trend at inner-city urban school yards: Having started in the mid 1970s in my town, even before discriminators, so thus having "seen the evolution" and slow working out of school yards in my city, I noticed that for school that were, say, blt. in the early 1920s, the vast bulk of the silver pulled from these schools always seemed to be '40s/50s stuff. And when teens/20's stuff was found, it was usually slicker (showing they were '40s/50s losses). At first, back in the '70s/80s, we kept writing it off to that we needed more depth. But as depth increased, and as an occasional early loss crisp teens wheat or barber were found at these '20s schools, we began to see that depth was not the issue. The earliest strata was definately within our depth range, but that the demographics of the losses from that earlier era, was not the same as it was in the more affluent times of the post WWII era.
One exception to the lack of earlier coins from 1-room school house sites, was if the school house site might have doubled for other purposes: Sometimes they were also used as grange halls, Sunday church services, community meetings, etc... which would bring adults.
By all means check ANY one room school house site. But this was/is just my experience. The best bet for shear numbers and quality of coin finds will be places where adults gathered and commercial ventures took place (coins/money changing hands, travelors coming and going, etc...). Like stage stops, saloons, camp sites where people slept in lean-toos, etc...
