urrm he used the hawk to catch naruto against kaguya so of course naruto knows about his hawk from then. hes not one to forget things concerning sasuke. so that is not proof that sasuke had sent him letters before. in fact that whole situation it looks like sasuke just recently returned to the earth dimension for the first time- its like people forget sasuke searches for this clues in a separate dimension. this which would explain why his doujutu was weakened.Then why compare Sasuke situation to them in 1st place?
Point is Sasuke has time to send info to Naruto(naruto recongized his hawk so it wasnt the 1st time). He could have just as easily contact his family. None of this happened becasuse of Kishis hard on for pointless drama.
yes, its most likely for drama purposes because sasuke showed he cared about them. theres no other reason.