I installed a front 2-inch hitch receiver on my F250. With it I carry a platform with a large tool box holding about 200 pounds of tools and related stuff. The box covers maybe half of the grill area and presents no discernable difference in cooling. The extra weight on the front end is compensated for by the hitch weight of the trailer. Serendipitously, it negates the need to transfer more weight to the front via weight transfer bars.
I don't know how much of your grill would be blocked by your yeti cooler. It could cover a lot of it if the cooler is too high. If so, you could probably use a dropped receiver on the rack to expose more of the grill. I bet you could get it done without affecting the cooling.
If I were you, I'd go for it.