I think it's because of what the mission is supposed to represent. The flag raising is filmed by a drone because it's propaganda. It's supposed to show the Helldivers conquering land deftly and confidently in their shiny badass armor, bristling with guns at the ready.

Filming a flag raising on a helldive mission as enormous swaths of enemies kill and throw the Helldivers around like basketballs is not something you wanna show as confident, powerful propaganda, so it doesn't logically make sense.