You can do that. But although such algorithms will produce correct answers to any NP problem when given correct answers to SAT, that does not mean that they will produce approximate answers to any NP problem when given approximate answers to SAT. (In fact, I'm not sure if the concept of an approximate answer makes sense for SAT, although of course you could pick a different NP-complete problem to reduce to.)

Edit: My argument only applies to algorithms that give approximate solutions, not to algorithms that give correct solutions with high probability, and reading your comment again, it looks like you may have been referring to the later. You are correct that if you have a polynomial-time algorithm to solve any NP-complete problem with high probability, then you can get a polynomial-time algorithm to solve any NP problem with high probability. Edit 2: sort of; see discussion below.