This has close to everything you want, although the house needs some updating:
https://www.redfin.com/zipcode/20815/filter/viewport=38.99157:38.95614:-77.06354:-77.11848.
It's also not super close to shops and metro, but walkable. Half acre though, which is very rare.
This also has a large lot and I think is probably quite walkable to downtown Bethesda:
https://www.redfin.com/zipcode/20815/filter/viewport=38.99157:38.95614:-77.06354:-77.11848
That said, if you can afford those, you could probably afford private school. Although maybe you're just someone who prefers public over private regardless of financial situation.
In general, the Kenwood neighborhood is best for larger lots and nice houses, but somewhat walkable (depending on where in the neighborhood). Caveat is that I don't think the middle school and high school are "10s" on Great Schools. They're still considered very good, but they are not 10s (fwiw, we're in that district and happy to be, so I'm not saying I don't think it's great, I'm just acknowledging that GS and others would argue the Pyle/Whitman tree is better).