The fact that "even the newer schools haven't been picked over" doesn't mean that older spots like the one room school house haven't been hit. Because quite frankly, more serious hunters skip new schools & clad. Hardcore hunters go out of their way to avoid clad, even to the point that if they're hunting old turf, they'll pass the shallow clad, trying to hone in only on deeper turf targets. That leaves a lot of clad, and the impression to some "hmmm, no one's ever hunted here before" So if there are any number of serious hunters in your area, and if the school's location is no secret to the history books, then odds are, it's been hit. I'd still try it though!
In my area, most of the one room school houses that had short life spans in the 1800s, are not that good for hunting. Jim in TN is right: Only if they served multiple purposes, like acting also as a grange hall, community room, gatherings, etc.... then the odds go up. But if it was strictly a school, the country kids had little need to carry coins in those years. I've hunted many that were virgin in my area. We knew they were virgin because the source info we got their locations from were from sources that had just been made known (original source history book just published). We'd slueth out the spots, find a foundation in the middle of nowhere, and go at it. Judging from the "woulda-couldas" (harmonica reeds, henry shells, shoe buckles, etc...) we'd know we were the first ones there. But coins were always scarce, if any, unless they had been used for other commercial concerns.
I noticed something else when we'd hit these virgin one room school houses: If, let's say, the school operated from 1895 to 1945, invariably, if a few coins were found, they'd always be at the later end of the date spectrum (20s to 40s wheats, etc...). We'd always assume "aha, then a barber or IH can't be far behind!". But usually, we'd end up with only the later dated coins. It began to be clear that the demographics of the United States economy changed drastically in the post WWII era. You know, like starting in the 1950s and up till today, EVERY kid has a few coins in their pocket (and perhaps dozens today) because we now had school lunch programs, affluent economies, etc... But in earlier times, things were different. I notice this phenomenom even at inner city schools. Having started detecting in the 1970s, we noticed that there was no shortage back in those virgin times, of '40s/50s loss silver at the schools. Even schools that were built in the teens and '20s we still found primarily '40s/50s loss stuff. At first, we just assumed it was because we weren't reaching deep enough. But when an occasional teens or 20s loss coins did get found, we could see that depth wasn't the issue (as they were still in easy depth grasp range). And newer deeper seeking technology in the '80s and '90s still did not produce the massive amounts of older silver, even though depth was doubled from earlier years. This shows that the post depression/post war economy greatly changed the amount of money that people carelessly carried around, especially for young children.