It's hard to wrap my mind around that a person cannot actually enjoy a game if it doesn't have arbitrary goals to work for that the game itself acknowledges with a notification. I don't hate it, but I find it extremely weird. There were whole generations where achievements just didn't exist. Hell, use Persona 4 as your example, as it was originally a PS2 game before it became a Vita game. Did these people with this opinion never play a game before the Xbox 360 or PS3 came out?
I can understand the idea of it being fun to work for side things. I cannot possibly understand a game being ruined because it didn't acknowledge that you worked towards a side thing.
Let's even go further with this idea. Games sometimes in themselves acknowledge that you went out of your way for something and reward you. Both Persona 3 and 4 do this by giving you something for maxing out all social links for example. However, this didn't reward you a trophy or something you can see out of game like the current points/trophy system does. Does that make it an inferior system capable of ruining your enjoyment?
