Ideally, yes.
In reality, it would take far too much time for each school to examine each site, report on it, discuss whether or not to block it, and then finally vote, then update the list.
Imagine how long it would take to vet 1000 sites...
Imagine in my town, with four elementary schools, a middle school and a high school.
The first rationalization would be to say "a single common blacklist for the elementary schools", rather than repeating the same job four times.
The next would be to say "if a site is on the high school blacklist, it goes automatically on the middle school blacklist; if it's on the middle school blacklist, it goes automatically on the elementary school blacklist".
Then, my town's schoolboard joins forces with two or three neighbouring towns, to keep a single set of lists for, let's say, eight elementary schools, four middle schools and four high schools.
Great, we've reduced the overhead from keeping sixteen separate lists to just three lists with a high degree of commonality.
So how about an open common project to maintain list? To block or allow a site is put to an online vote of parents (or combination of parents and school principals, maybe district superintendants). This is bound to provoke "lively discussion"...
K.