They definitely do. I read an article about it some time ago, but can’t find it atm. They’ve funded hundreds of movies even just since 2000. Not sure about the actual monetary support, but just providing equipment really is an invaluable asset for someone making a movie with or about the military. Naturally the US military wouldn’t be keen on supporting projects critical of them, but more than keen supporting projects portraying them in a positive light.

On Godzilla - only watched the first, but yeah…same story as Transformers. Those two and superhero movies are definitely some of the more insidious ways they push that line. Like ok, Black Hawk Down is hurra military bs, that’s obvious. But eg Transformers is almost more vulgar and open with its “Yay US military” line, but so very thinly veiled by cool big car robot fight for kids.