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PB is pretty good - but only the natural kind. Brands like JIF and cheaper brands add sugar and use "bad" oils along with tons of unnecessary preservatives and additives. You won't find this in your school cafeteria.
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While I agree with the rest of your post, I don't think I can agree with the statement above. I don't see how a 'natural kind' would be much better than something like JIF. The statement about 'tons of unnecessary preservatives and additives' is just incorrect. There are no preservatives in JIF (not that they would be necessarily a bad thing) and no additives aside from simple fats, and even the 'bad' oil is less than 2%, and it's more like a useless type of fat because it's been partially hydrogenated to give a better consistency rather than, say, being toxic, which would be bad if it was the bulk of the fat you consume but again it's just less than 2% of a food that's high in otherwise mostly essential fats.
Also, in terms of sugar, granted I am not an expert in PB since I don't particularly like it to begin with, except as part of a few types of chocolates, but I am not sure how PB would taste without any sugar or what would be the point..
I mean at that point it seems like you would be much better-off with eating roasted peanuts directly which at least are delicious as they are... especially in terms of eating something that tastes good.
Maybe there are people who would still like it without sugar, but I am not sure I would even call it PB at that point, more like 'pre-chewed peanuts with oil added" (yuck). It sounds as weird as recommending people who like chocolate to replace it with pure bitter 100% cocoa powder with nothing added, which tastes quite different (more like sand).
(for reference: JIF PB Ingredients "MADE FROM ROASTED PEANUTS AND SUGAR. CONTAINS 2 PERCENT OR LESS OF: MOLASSES, PARTIALLY HYDROGENATED VEGETABLE OIL (SOYBEAN), FULLY HYDROGENATED VEGETABLE OILS (RAPESEED AND SOYBEAN), MONO- AND DIGLYCERIDES (simple fats that become triglicerides a.k.a even more normal fats in the body, here they help with consistency) AND SALT.")
So again, I would also say that you should think of a PB+Jelly sandwich as the equivalent of a nice piece of cake with lots of frosting (maybe a bit like cheesecake in terms of the bit of extra protein from the peanuts) and not as an actual meal or normal nutrition.
And at that point I really don't think you should worry about any difference between a cheap standard brand and some other brand that claims to be 'all natural', because really there won't be any appreciable difference on your body between them (well unless it was sugar-free, but then again if you cared about sugar-free you shouldn't eat the white bread and jelly either and should stick with normal roasted peanuts).