There's no evidence for this theory at all. Humans have evolved to eat just about anything, which is one of the reasons why we became such a widespread species.
That reads quite harshly, I'm not sure if your dead set against it or merely suggesting another interpretation. Personally, I'm not convinced either way with the 'evolutionary perspective' argument but I think it one that should be considered.
I think your wording ("humans have evolved to") might confuse the argument a little. Strictly speaking, humans don't evolve to do anything specifically. We evolve randomly and natural selection sorts out the survival aspect (though arguably, today, modern medicine helps us out there too!). Its follow then, to me at least, that there is a high probability that we have, in the past, eaten stuff that has killed us! It's our evolution that has allowed us to survive on what we have eaten.... which is almost the reverse of your wording. Moot point perhaps but I think important in understanding evolution and natural selection.
With that context in mind, it follows (again in my mind) that new food stuffs in our diet have an as yet unknown long term effect on us. We haven't in the past eaten the additives and chemicals that form a part of processed foods these days. And these are invented/discovered (and added to foods) much much more quickly than we evolve! Btw, when I say 'new food stuffs', 'long term' and 'in the past', I'm using the evolutionary time scale... millions of years.
I think your point about 'new' diseases being attributed to our longer life spans is valid. Though interestingly it can be understood using the evolutionary perspective too; that is, we're in uncharted waters - we simply haven't lived this long before so haven't evolved down this line! Give us another couple of million years and we could well be living disease free until we're 150!
To be perfectly honest, I don't know what the answer is. I haven't even decided which notion I side with (i.e. 'we can eat anything' or 'we have not evolved to eat the new stuff') but I don't dismiss either out of hand. What I do go by though is the equivalent of the mirror test for weight loss; if something I eat disagrees with me or just makes me feel bloated and farty, then I limit my intake of it! Be that white bread or a fresh home grown apple.