It frequently, but not always, plays a role. I know it was a factor in me choosing Blue, Leaf Green, Pearl, and Y, and maybe Silver and Shield as well - I don't actually recall for those games, lol
Most of the time, I'm looking for the version that includes Pokemon I want to use on my main in-game team, though for a couple games, I made my decision based on which game didn't have certain Pokemon that I don't like.
The box legendary exclusive, however, doesn't factor into my decision since I just box those anyway.