Supermarkets where I come from do check characteristics of ingridients like pesticide content. They generally care about providing quality products.

If a supermarket wouldn't do quality management of their suppliers I would consider that bad.

That is meaningless unless

1. "quality management" can refer to arbitrary characteristics, in which case, no, most supermarkets will not keep track of whether the wheat was harvested on Tuesdays, or
2. You're assuming that there is something special about GMO such that it counts as "quality management" while whether the wheat was harvested on Tuesdays doesn't.

Information provision is not about whether people should care about it but whether they do. In this case plenty of people do care about.

I'm pretty sure plenty of people care whether the produce is picked by illegal immigrants, at least to the extent that if they're told, it would influence their decision. I'm also pretty sure people would care if the company owner is gay, or has had an abortion, or any of a number of politically charged things that we don't demand should go on labels.

Don't make fictional examples that wouldn't work in reality in the first place, if you want to train reality based reflexes

There's a difference between not working for reasons that affect the point and not working for reasons that don't. The example is of a politically charged trait. If one politically charged trait isn't workable, pretend I instead mentioned another that is.

If you don't think abortion is a good example, change it to "has been disclosed as a campaign donor to a politician of party X" or "has refused to take an IQ test/has tested at an IQ of _" or whatever politically charged example you think is valid.