Does using egg+yolk in a chocolate chip cookie recipe make a big difference?
The main reason that you'll see recipes use a specific part of the egg is that the whites and yolks have a very different composition.  The whites contain a large number of structural proteins (which is why they can be whipped into stable foams) while the yolks contain most of the fats in the egg. Fats and proteins play very different roles in baking.

So, if you see a recipe that uses a whole egg plus an additional yolk, the yolk is there to add fats without adding the additional proteins that the white would bring to the party. This affects the structure of the batter and the texture/flavor of the finished product. Generally the extra fat will contribute extra softness and richness, from its composition and of course extra moisture. Here's a good question with related information.

In regards to commercial production, I'd be really doubtful that large commercial producers are using anything like whole eggs. Based on scale alone, they're more likely to be using pre-separated eggs from a supplier, or perhaps some kind of horrifying egg substitute. Not to mention all sorts of stabilizers, emulsifiers, preservatives, and so on that all perform functions similar or complementary to the chemical action of the egg.

As for extra whites, there's any number of uses for those. Powdered egg white is a surprisingly common food additive because of its thickening and emulsifying properties, and it's possible to find packaged, pre-separated egg whites in cartons. The same producers that make these are likely supplying the commercial baking operations too. Don't go imagining that the Chips Ahoy factory is full of a bunch of enterprising bakers cracking and separating eggs all day long, casually chucking out extra whites. If there's waste, it's further up the supply chain, and most likely the supplier is finding something to do with those extra whites.