But why 3 steps of compilation?
The first step merely shows that you wrote valid C code that gcc can compile. It doesn't prove that the program actually does what promised. For example, if you missed to implement 'for' loops, this step would still produce a compiler, which could still work for a subset of C.
Here comes the second step: it proves that it implements enough features to at least recompile itself. Now, this is still not a guarantee that all C constructs and features work, but we can reason that even if it only implements a subset of C, it's a large enough subset to build such a substantial app as a compiler. How likely is it that a compiler doesn't use a 'for' loop, not even once?
But what is the reason for the third step? It doesn't add anything, it doesn't trigger any code path that was excluded in the first one. After all, if you did miss 'for' loops, and the 2nd step hasn't detected it, it must be because there are no 'for' loops in the source, so you will never detect it this way, no matter how many steps of recompilation you run.