As for the remark at the end: you're right, it may be difficult to un-obfuscate phrased in that form, but had you phrased it "Make a table of two columns. Take the very top-most, left-most entry to be n, take every entry in the right column to be 1 plus the value to its left, and take every further entry in the left column to be the product of all previous entries in the right column. I claim the value three cells below the initial n has three prime factors (and similarly the value below that has four prime factors, etc.)", and I couldn't see why that would be the case, then, yes, this would give some real pause as to whether I could truly claim sound understanding (to the point of considering it obvious) of the reason for the infinitude of primes.
And if, analogously, the obstacle to someone seeing "the smallest positive value of the form 98X - 60Y is 2" is merely inability to factorize 98 or 60, well, that doesn't mean much. But I don't think that's the obstacle for most; I don't think people would do much better were it phrased "What is the smallest value of the form 2 * 7^2 * X - 2 * 3 * 5 * Y? (BTW, 2, 3, 5, and 7 are all prime)". And I do think, while not definitive in itself, the fact that this sort of thing is not obvious at all to people suggests we shouldn't presume sound reasoning behind uniqueness of prime factorization/Euclid's lemma/etc. is implicitly obvious either. When people claim it to be obvious, it's not because they have a dim view of the correct reasoning, seen through a glass, darkly. It's just because they're making complete logical leaps.
And if, analogously, the obstacle to someone seeing "the smallest positive value of the form 98X - 60Y is 2" is merely inability to factorize 98 or 60, well, that doesn't mean much. But I don't think that's the obstacle for most; I don't think people would do much better were it phrased "What is the smallest value of the form 2 * 7^2 * X - 2 * 3 * 5 * Y? (BTW, 2, 3, 5, and 7 are all prime)". And I do think, while not definitive in itself, the fact that this sort of thing is not obvious at all to people suggests we shouldn't presume sound reasoning behind uniqueness of prime factorization/Euclid's lemma/etc. is implicitly obvious either. When people claim it to be obvious, it's not because they have a dim view of the correct reasoning, seen through a glass, darkly. It's just because they're making complete logical leaps.