But it's rationally unsound and unscientific to argue that all GMOs are unsafe and evil. So if anti-GMO people want to stop people from eating/producing GMOs, they should find less opposition to specific labeling than to generic labeling. Otherwise it's too easy to paint them as uninformed.

I think you're right that the intended purpose of generic GMO labels is to muddy the issue and spread uncertainty, which is exactly what they do. I am arguing that this strategy is counterproductive in the long-term, both to the anti-GMO cause and to society's health and informational well-being.