Turning stone to cobblestone

Not sure if there is a specific separate system to allow for that conversion.

The only way you would be able to do that would be to place the stone back down and mine it.

Oh, there are ways. You can run it through a blast chamber. It's a good way to convert a couple (or a couple dozen) chests of stone into cobblestone, if you happen to have it. Makes very little sense with a stack or two (more is needed to fill up the conveyors between the input and the blasting area.)

You can manually place the stone, TNT and blow it up. Not very efficient and rather costly. You can bait a creeper into detonating it - most will be destroyed but you'll gain a couple blocks. Or if you feel daring, you can bait a Wither into breaking as much stone into cobble as you like.

But practically? Stash that stone for use when you need it, and either grab a pickaxe that isn't enchanted with Silk Touch and dig up some stone from under ground, or if you feel ambitious and need LOTS of cobble, build a cobble generator. It's a contraption that can produce hundreds of thousands cobblestone blocks per hour out of nothing (or more accurately, out of flowing water that can be replenished infinitely, converted to stone by flowing lava.)