I wonder why they didn't just use a battery ?

Because batteries have to be replaced, requiring opening the player. And for the most common usage of the player, the capacitor should be more than plenty.

Keep in mind that if your player's clock isn't keeping time, there's a malfunction. It's got nothing to do with their basic design decisions. Personally, I'm glad there's no battery inside my player that I have to worry about.
_________________________
Tony Fabris