Branding food as GMO has only negative connotations. I think the way forward is to demand for a non-GMO label. This is more realistic and will work better when partially implemented.
It makes zero sense for companies to be required to list what is NOT in their food. Yeah, let's scrap listing ingredients and instead list everything that isn't in an item. ;)
Non-GMO is not an ingredient, it is a classification. Ingredients either classify or not and the label on the package speaks to the classification. A common example is "not from concentrate".
Edit: (sorry for changing my comment, I add the original below).
You are picking on the wording. There must be a succinct way to say non-genetically modified animal.
I'm making an analogy to the earlier remark that labeling things "has GMO" is like labeling them "has excrement" and "is radioactive."
"Has excrement" would certainly be perceived as negative, but the effect on my shopping of "has no excrement" would be a lot more negative (trying to find this sticker on everything I purchase, for example).