The only big website that could even come close to doing this (they won’t, and if they did it wouldn’t work, but they’re big enough that the attempt would at least be noticed) is Wikipedia.
A slightly more “productive” (sort of) avenue of approach would be another large corporation for whom Google is a competitor, and who themselves doesn’t rely (as much) on advertising, interfering with WEI for their own self-interested reasons. Apple is the most likely candidate here, although again, I don’t think that’s likely to happen.