I don't see how that makes economic sense. Preserved food should wind up cheaper than the highly perishable even with additional input costs barring truly massive differences - the surplus isn't lost and can build up and even if it the sale price minus processing costs is less per item something is more than nothing.
As for your grandparents they were poor farmers so of course labor and seeds were cheap for them - they already had the prerequisites by definition.
I'm really befuddled at some of the reasoning here.
You buy groceries, right?
Because if you do, and you look at the prices, well then it's pretty clear.
So this should be an exercise in understanding why that is the case, not trying to reason against the reality that produce is cheap.
Most produce, and most of the best foods are really cheap.
Even other things like lentils, beans, quinoa (a little more expensive), chick peas.
You just have to cook it.
As for your argument 'preserved foods should be cheaper' - which by the way is a reasonable point for someone who takes an academic perspective and doesn't actually buy groceries, the answer is simply that those 'massive costs' are indeed that massive. They include expensive things like 'metal' in the cans. Fuel for transportation. Branding. Marketing. Distribution. Supply chains. Etc.
Fresh food is accessible to everyone in America with any kind of domicile. The notion that poor people eat more processed foods due to economic situation doesn't hold a lot of water, though there might be some issues around the margins (again: time), but I suggest it's more simply due to bad behaviours and bad eating habits.
FYI: poor people are also much more likely to smoke - which is a prohibitively expensive activity. Granted - it's addictive - but still.
The thought of homemade garlic hummus with baked peppers has inspired me to go and do that literally right now. It'll cost me $1 ... but a lot of time which I don't really have ...
Edit: My grandparents did not have electricity or cars as children and did not have plumbing until middle age w/ several children. Does that count as 'poor'?
As for your grandparents they were poor farmers so of course labor and seeds were cheap for them - they already had the prerequisites by definition.