I grew a ton of potatoes in a non hilled space. It was a 12 foot deep square foot garden. I did nothing to them but water.
The problem with the square foot gardening method is that it requires extremely high amounts of organic matter to be added yearly in the form of compost and rotted manure (up to a third of the volume of the garden bed), plus other amendments when setting up such as vermiculite and peat moss. Add that to that fact that they dry out extremely quickly (meaning they need
lots of irrigation). In my opinion, square foot gardening is about as far from being a viable SHTF gardening method as it is possible to get.
That said, I don't think growing potatoes is really that labour intensive. Granted, we only grew 20 square meters (about 200 square feet) but we grew it on land that was previously sod and nettles. It took a couple of hours to dig the bed, three sessions of hilling (1.5 - 2 hours) and probably a few more hours of digging, cleaning and sorting. We mulched a few times with grass cuttings and added some soil amendments like composted manure and now we have a really nice garden bed that is deeply dug and fertile, ready next year to be planted with legumes. The main problem with potatoes is that if you have aphids your seed potatoes could eventually end up full of disease and growing very poorly. I've been hoping to try growing potatoes from true seed but every variety we have tried produced flowers which then dropped off before setting fruit (no, not a pollination problem).
The original question is a tough one, as living in Sweden limits what we can grow successfully. Plus my DF really hates beans. I assume I am living in the same place which means I still have access to our fruit trees and bushes and am able to forage in the forests nearby which contain plenty of mushrooms and edible berries in the autumn. That said, I'd probably grow peas, potatoes and either swede (rutabaga) or beet. All three can be stored easily without power.
My selection would change depending on the circumstances of where I am living and whether I have to bug out or in.