But I disagree that laypeople are correct to consider unique prime factorization (or Euclid's lemma, or any such thing) for integers obvious, or have correct reasoning for it latent in their head.
You gave for example a perfectly correct argument in https://news.ycombinator.com/item?id=11957549. I disagree that laypeople have anything like that in mind when they are claiming these facts to be obvious.
The process that leads laypeople to consider these facts obvious is not based on their having any special intuitive understanding of the multiplicative structure of the integers. It's just the process of "I've never seen it fail, and I keep hearing it's true, so... yeah, how could it be otherwise? How COULD it be otherwise?", the same process that leads them awry in other cases where they draw actually wrong conclusions.