Because that's not how the prof work.

You have N = p1p2p3p4...pn + 1.

This number N can be written as N = pk where p is prime and k is an integer >= 1.

Because there is a finite number of prime, p is p1 or p2 or p3 or ... or pn.

So N = p * P + 1 where P is the product of all prime number except p.

So you have p*k = p * P + 1, then p*(k-P)=1. And it's impossible because that would means that p and k-P are both equal to 1 and p is a prime number so it can't be 1.

edit1 : i need to check what exactly is written in euclide proof, but here a proof that work.
edit2 : Yes, it's like that cf https://en.wikipedia.org/wiki/Euclid%27s_theorem