When did 方便面 become 泡面 and why?
I think, your intuition is right. 泡面 sounds fancier, and people prefer fancier names unconsciously.

方便面 sounds cheap; they were just instant noodles for filling one‘s empty stomach. The name reminds the way how migrant rural workers eat 方便面, which is very common to see in train stations of big cities.

泡面 sounds as elegant as 泡茶 (make tea); because usually it takes time to 泡 (verb) something and then to enjoy what has been 泡-ed.