Brewing with very bad old-smelling potatoes?

Buy fresh potatoes regardless of what you want to do with them. Come on!

I'm deliberately avoiding addressing whether this question is appropriate for this forum. A question was posted on meta regarding that issue, presumably in response to this question, so it's possible that this will still be closed. In the meantime, however...

You could brew beer with potatoes, and there would, of course, be challenges in that alone - namely ensuring that you have the enzymes to properly mash the potatoes. Pun intended.

What you do with this potato beer after brewing is your own business, and probably outside of the scope of this forum.

More to the point: you should not brew with spoiled potatoes. It's just not a good idea. I would think the god-awful smell of heating rotten potatoes would be reason enough, but there's also the risk of fungal toxins like trichothecene, and no way of knowing if any toxins are present. Even if your potatoes weren't toxic, I'm not sure that you could ever get the putrid smell and taste of rotten potatoes out of the brew. Even with extensive "processing" after brewing.

You might need to rephrase your question to better align with this answer and focus solely on brewing with potatoes, since that's what this forum is meant to address.

But seriously, throw the potatoes away.