1. " If you think it’s obvious, then you’re probably assuming what you need to prove" i.e. "If you think this, you're wrong."
2. "Just because you’ve got a completely deterministic method for working out a prime factorization, that doesn’t mean what you work out is the only prime factorization" i.e. "If you think doing it this way works, it doesn't."
3. "Look, it just bloody well isn’t obvious, OK?" i.e. "If you get frustrated that I keep telling you you're wrong, that's your fault for being wrong."
4. "If it’s so obvious that every number has a unique factorization, then why is the corresponding statement false in a similar context?" i.e. "I won't tell you what you did wrong, but instead I'll show you why your answer can't possibly be right."
It is unconscionable that he does not identify the actual fact people erroneously use without justification when thinking the fundamental theorem is obvious. FYI, that fact is
"If a number n can be written as the product of a fixed list of (not necessarily distinct) primes p_1, p_2, ..., p_n, then any prime p dividing n appears on this list."
(there is a minority of mathematicians who refer to this fact as the fundamental theorem of arithmetic, not unique prime factorization; the obvious argument is going from the fact to unique prime factorization, proving the fact is the unobvious argument)