No they don't. Here in Norway the bottles are compacted in the back of the reverse vending machine (usually from Tomra: https://www.tomra.com/nb-no/collection/reverse-vending) and sent directly to recycling. Bottles from all the different suppliers go together, none of them go back to the manufacturer of the content. And Coca-Cola has a deposit on bottles here and it is an entirely non-controversial idea.
Moving to the system presently in Norway, and Germany I believe would be a huge step in the right direction. Sadly most governments appear afraid or unable to actually put accountability on corporations. Personally I'd require them to go back to deposit and reuse.
These are the biggest beer brands (Carlsberg etc) and the biggest soda brands (Coke etc), though the glass soda bottles are mostly seen in bars.
The thing is, it does make the product more expensive. A ten cent deposit on a dollar product means a customer has pull $1.10 out of their pocket or CC balance. Customers don't think logically, and even if they do, they have to assign some monetary value to having to bring the bottle back, the chance that it may be discarded, etc.
It does hurt sales. If it didn't, the Coke would already be $1.10. (My exact numbers are fictitious, obviously, but the point is that Coke has already set their price at the number that maximizes profit.)
I'm not opposed to bottle deposits. They're highly effective. I just think it's disingenuous to act as if Coke here is opposed to recycling in general. They're not. They're opposed to the reduced sales that will come from higher prices.
That difference gives recycling advocates options. They can agree to spend some money on educating consumers, or making bottle returns easier, or anything else that might help ease the effect Coke is trying to avoid. They can probably find some common ground. There's no possible reason for Coke to be opposed to recycling in general.
Do you have any evidence for this?
It seems like any fee, even a small one, is going to impact sales when an item is priced as low as soda is. That's not to say their stance is justified or morally right. But framing it as primarily driven by opposition to recycling (which is a source of cheaper raw materials for them), as opposed to blanket opposition to any kind of fee being added to the price of their products, strikes me as misleading.
So the question becomes why do they oppose a fee that is unlikely to hurt overall sales? Ah, because they would have to do something with those returned bottles...