Hello Edlee

This is merely my educated guess,
but I suspect there are generally three groups of people
when it comes to oiling the slate stones: those who prefer them oiled, those who don't, and those who don't notice or don't care.
Those are the only three possibilities...as far as I can fathom

So my question is: since you mention the most important part before using a nice set of slate and shell stones is to wash your hands to remove and dirt or oil from your hands (which I agree 100%),
then it seems to me quite counter-productive and counter-intuitive to intentionally put mineral oil on the slate stones,
because during play, the net result is you are constantly transferring the mineral oil, however thin its layer, onto the shell stones.
And not only to the shell stones, but actually to all the other surfaces: the Go board, the bowls, even any books you touch after you pick up any oiled slate stone. Food, even. In effect, you're smearing and spreading mineral oil everywhere!

This part confuses me a little. Thanks.
New slate stones actually come with mineral oil already on them. It's recommended you wipe them down a couple of times before play.
My take on it is that the oiling after cleaning doesn't serve a purpose other than to give the slate a newer look. In truth, the more you play with the slate, the more the moisture from your hands will do that job for you so it's a matter of aesthetics.
As for it being counter-intuitive, you kinda got me there

. I wish I could give you a good, technical answer, but I can't. I suggested wiping them down at least three times just to be sure you removed the excess. Properly cleaned, I've never seen the oil smear the board or other stones.
I haven't looked at the Go Players Almanac for almost three years, and it has some useful information on the whys. I'm not home at the moment, but I'll check this evening and see if it provides any incites.
If you wanted to oil your slate stones but wanted to avoid using mineral oil, there are a few good choices. 100% pure camellia oil would be a great alternative. It's used on knives and other metal equipment in Japan. Sadly the health foods craze in the west has latched onto it and it's difficult to get some that isn't intended as a hair or skin product.
I'll probably get some flak for this but what I use (to great effect) is pledge orange oil (for those not in the U.S. it's used to revitalize wooden furniture). It makes the stones look great, whatever's left over after you've wiped it dries, and the orange smell will fade after a few days. I didn't include this in the article simply because it's not part of the norm.
Whatever the case, I recommend avoiding most vegetable based oils (especially the heavy stuff like linseed).
That's probably more of an answer than you were looking for Edlee, sorry for going off on a number of tangents
Avoid cleaning slate stones in a metal container because the black slate is slightly abrasive and metal can be rubbed onto the stones. This is especially likely to happen with aluminum. I experienced this when I used an aluminum colander to rinse off the stones after washing. The black stones had metalic stripes on them

Fortunately this could be removed by serious scrubbing the affected stones with a plastic scrubbing pad but took a lot of work.
TRUTH!!! I need to include this in the article.
I did something similar with one of the first used sets. It left terrible streaks that took forever to remove. I didn't want to sell the set afterward so I kept it for travel and Go Clubs.