I think it's 2 conflicting things going on. An owned brand/private label has higher % gross profit, but a name brand will often have a higher $ gross profit per unit. There's a balance for total profit between the whole set of products, because some people are brand loyal and others will switch depending on the price difference and options.

Where I work, store brand gives us a huge margin, even at a low sale price. There’s not a single middleman trying to extract a profit.

So why is the store brand not in the prime spot rather than the bottom shelf. Is the major brand paying for placement?

Yes, it’s called a slotting fee.

https://en.m.wikipedia.org/wiki/Slotting_fee

Clasic economics would say therefore the lower profit margin + slotting fee should equal the higher margin of the house brands.

I wonder if that's roughly true? I could imagine both yes and no. Classic economics, sort of by definition, can't take real world edge effects into account.