What would a balanced output do for the 3 feet of cable that will most likely be connected to it? I mean sure, put it in if you can't help it, but even though it's a portable player, no one is gonna take that to the next motor fab where it would benefit from a balanced output, and at home an audiophile most likely already has a better player around.
And yes you're right - the whole idea is nonsense to begin with. CDs have always been fully digital, so better listening options exist.
A balanced output will have less crosstalk between the channels. I've never used balanced headphones, so I don't know if it's noticeable. My guess is the only really noticeable thing is that the higher output voltage swing from the differential amplifiers will make high impedance headphones louder.
I have a bunch of audio equipment in a studio and do run some balanced lines to stuff that requires longer cable lengths that are in proximity of unrelated, noisy devices (e. g. digital stuff).
Can't say I can tell any difference for headphones.