The places I know were they do cook stuff using volcanic heat (in Peru and the Azores islands which are part of Portugal) they do it by digging a hole in an area were the ground is hot from volcanic heat and putting a pan cooking in it (they cover it all to keep the heat).

So it's more a local technique for cooking for free that then evolved into a couple of traditional dishes.

Never heard of trying to roast stuff on the output of a geyser.

Would depend on the specific hot spring. Most would cook and dissolve it. Additionally it would be very Sulphur smelling and tasting which would be range from icky to deadly depending on how much of the undissolved you ate.

Does this actually work to coock it?

Doesn't seem safe to try to get your cock in it while it's in the hot spring. Maybe you could find a way to make it could actually work...but why? Are you a masochist or something?

Is it at all edible?

Sous vide method would probably have the most chance Of being edible since the turkey would be vacuum sealed

Is there any environmental impact or downside?

Yes. That's why the park service is saying not to do this. You'd be introducing new chemicals into a delicate ecosystem and also potentially physically damaging it.

It would cook it, the springs are hot and acidic enough. You’d just have to sit for a long while. Edibility depends on your allergies and tolerance for poisons.