Here's my imo.
standing from a mass production point of view, I dont see it as cost effective. I can see they would have no problem producing mass quantities, but I cant see it being satisfactory. Companies would have to employ a way to keep mtn dew ice cubes frozen during transit, which would invoke a new cost they never had before. If you've ever seen a soda vendor restock trucks, theyre actually kept at room temperature. To pay for the extra refrigeration, they would have to drive up the cost of their soda. After all, it takes way less gas to keep a truck at room temperature than keep it below zero degrees. The other option would be to produce the ice at the place of dining. It should not impact labor cost too heavily, but then the facility has to have of means of keeping up with demand during rush. They also would have a hard time storing it . ice is prone to absorbing flavors of stuff it's stored with.
When a carbonated beverage freezes, it causes the co2 to dissipate. So when it melts into your drink, it will cause it to go flat much more quickly. But I dont see that being a problem for say, hawaiian punch or lemonade.
Tl;dr expensive massively, fun at home.