"When thinking about who we are we must remember that America was discovered by Europeans while on the search for food."
I think there is a mistake in this sentence. We say "in search of". Do we ever say "in the search of" NO or "on the search for"NO(?) ? I doubt them being correct.
I suspect that the writer has conflated 'in search of' and 'on the hunt for'. I have put a question mark after my second NO above, because I feel we may use it. It is not a natural idiom to me, but it doesn't sound totally unnatural either.
I haven't any reference boks to hand to check, but your "I doubt them being correct" seems unnatural to me. I would say, "I doubt that/if/whether they are correct".
Comments from others on both points would be appreciated.