I get the feeling that if someone understands how to square x + dx and can also follow the similar triangle argument on the next page, they'll be totally fine studying calculus starting with limits instead of this.
I find the geometric proof of the product rule using differentials much more intuitive than the difference quotient proof. The difference quotient proof is a clever algebraic trick, but it doesn't (at least for me) give any deep insight about why the product rule works.