"I wonder if 'taters can be left to go wild like that?"

Kinda-sorta... any little 'taters you missed during harvesting will probably sprout the following spring, although that may depend on how hard your winters are. If any sprout in spring, they also tend to be green with solanine, so you can't eat those, but green ones may be just fine as seed potatoes. (They may not be green if they are deep enough, but my chickens always uncover mind.)

If you deliberately left some smaller ones there, marked the edges of the bed, and sprinkled the area with a complete organic fertilizer in spring (no lime or ashes!), and then covered the bed with 3" or so of collected leaves or fluffed straw, adding more as the tops got taller, you might just have a decent harvest.

I think the big issue is disease -- the longer potatoes grow in the same place, the greater the chance for disease. If you started out with certified disease-free seed stock, you might postpone it, but potatoes have the reputation of being disease magnets.

This just a theory I would like to try someday, AFTER I have tightened up the interior fencing so the chickens stay out of the area. Unless a disease gets started in the soil from repeated use, it should work.

Right now, I just set aside the small potatoes at harvest and put them in an airy container and just let them sprout in spring, and put them in a new bed. This will be the third year for the same potato stock, so I'll see how it goes.

Sue