My school wasn't English, or as elitist as Eton, but each year on Remembrance Day they would read the names of every pupil who died in the two world wars. Took quite a while.
This is actually a reason why it's common. The impact was so large and so universally felt that nobody would object to the use of public funds to memorialize it. It merely takes someone to suggest it, and nobody would oppose.
Not sure about that. Certainly after WWI virtually every village raised some kind of memorial (even the Thankful Villages). In that environment I can't imagine a public school not doing the same. Even my comprehensive had a Roll of Honour on the wall (now I'm wondering what happened to it when they demolished that building).