SPUI is, essentially, correct. What this proof establishes is that there is no rational number a/b such that (a/b)^2 = 2. So, if the square root of 2 exists, it must be irrational. In a theoretical sense, there is no reason to assume that the square root of 2 must exist, and there are certainly plenty of examples in abstract algebra of algebraic structures in which square roots and other radicals do not exist. However, the Pythagoreans had a very practical reason for wanting to discover the square root of 2. Their math was mostly geometry, and as anyone who knows the Pythagorean Theorem is aware, sqare roots are an important part of geometry. So, with tongue resting lightly in cheek, the result could rephrased thus:
Any system of numbers complete enough to measure the hypotenuse of a triangle is going to have irrational numbers in it.
Since mathematics would be very limited without the ability to measure the lengths of things, what we really have is the proof of the necessity of irrational numbers.