Clearly, the sticky sticks are the result of the ageing process of the rubber used. I was recently decommissioning a Toshiba lappy from the '90s and the trackpad rubber "nub" in the middle of the keyboard had turned to sticky goo. A similar thing happened to the rubber on the latch of a 2004 Acer laptop I still use sometimes. So a little slime is not so bad, but they really should use some better materials. 6-7 years is definitely too short a time for stuff to literally start falling apart.
I did wipe the sticks on the DS, but alas they kept bleeding the slime. Haven't looked at them in some years now, so I can't tell how badly they've deteriorated since.
That said, I'm very proud of owning a fully functioning PSOne, even if it's a late model from around 2001.