Sure, but the label tells us nothing unless it identifies the particular strain(s) that have been modified. Simple "GMO" vs. "non-GMO" labels are anti-informative. They have net negative utility as they don't tell us whether something is safe or dangerous, they just muddy a discussion that could be perfectly clear.

IMO anti-GMO advocates should be pushing for more specific strain/cultivar labels on food, not just a useless label that says "produced with genetic engineering." Then pro-GMO, tentatively accepting, and anti-GMO groups can all be satisfied.

I submit that the "GMO"/"GMO-free" labeling serves precisely the intended purpose. If you believe that all GMOs are unsafe and evil, then what more could people possibly need to know?

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.

99.9% of consumers would have no idea what to do with that information and most products would contain several strains of each ingredient anyway.

99.9% of consumers don't know what to do with current ingredient lists except be vaguely confused by the long list of scary-sounding chemicals. We should probably still keep those.

Those ingredient lists are often used by people with allergies. So, they are useful to a wide swath of people.