You made an error: p1p2...pn + 1 is not necessarily prime, but none of its prime factors are in {p1, ..., pn}. For example, 15 = 2 * 7 + 1 is not prime, but neither 3 nor 5 are in {2, 7}. I agree with your overall point though.
2*3*5*7*11*13 + 1 = 30031 = 59*509