Ok I will give it a shot. There are two questions:
1) Do the new potatoes only grow from the roots below the seed potato or do potatoes form along the main stem that grows upwards?
2) If potatoes do grow off of the main stem, is there a window of time that the main stem needs to be covered with soil before it decides that it is going to root? I feel like I only get potatoes below and parallel to the seed potato that was planted.
New potatoes only form from the Nodules that form on the root system. The potato tuber that you harvest develop from these. So I guess to answer your question the new potatoes form under the plant near the root system.
But It is good to pull soil up around your plant as it grows. This gives more soil for the plants to develop tubers in.
This has nothing to do with your question but here are two ways to grow potatoes that sound a bit crazy but are supposed to work
1. To use old tires to plant in and as the potato plant matures keep adding tires and put new soil in the tire [keeping the top of the plant above the soil] The theory is you will have a bumper crop of potatoes develop in this additional soil and all you do to harvest your spuds is to lift tires up and there they are.
2. Dig a trench in the garden and lay a layer of straw down and then put your seed potatoes on this and cover with straw and build this up. The theory here is that you can harvest spuds throughout the growing season and the plants will continue to grow and the new tubers left will continue to develop.
I must say that I have not used either of these methods but both have been suggested from good gardeners who have used them and it has worked for them.
I still plant in the garden in the soil as I have for several years and have been successful with this normal method.