The rotation rule is a good rule. But I do not think one potato will over turn the tomato cart, so to speak.
The big problem is nematodes & the lack of nutrient.
I do not think one potato will make a different. You can plant the Nightshade family side by side in any given season.
Just not season after season, for many years on the same plot.The exception to the rule is that hot pepper & sweet pepper will cross & you will have hot sweet peppers.I know this works, it happen to me after 35 years of gardening.
The only other thing about planting the Nightshade(tomatoes,potatoes,peppers,eggplants,tomatillo)family together,is the horn worm & other insects will be able to find the plants easier in a big group.
Some companion plants: carrots,dill,garlic,marigold,onions,mint,beans are good.

I always thought that you could throw a whole potato in the compost bin and it would just add the same as anything else.

I found out to my tired fingers, this is not so, you throw an eye in there and it produces a potato. Throw some skins in there--with hidden eyes, and they do the same thing.

When we lived in the city we used to go to a city-run filtration centre where they used to put out compost for anybody who wanted to take it. Huge amounts of compost could be had free. It came from what all the waste they used at the facility including human stuff. The piles were given names..'good', 'better' and 'best'. Many piles of each.
There was also a guarantee that the 'best' pile was good for whatever gardening task the taker wished to use it for.
There was also a kind reminder that tomto seeds are most likely found in the piles and the taker should be aware of that fact.
They were so right.