It is not used literally everywhere. It does not always "just work," as the original post demonstrates. It often happens to be good enough for most practical purposes, yes, but arbitrary-precision arithmetic is better for most practical purposes.
Fixed-precision arithmetic has one main advantage over arbitrary-precision arithmetic: it is more time- and space-efficient. This advantage only applies if the fixed-precision arithmetic is actually correct and the fixed-precision arithmetic meets some concrete time or space constraint which arbitrary-precision arithmetic fails to meet. It generally takes time and effort to demonstrate that these conditions hold; because one can rely on the correctness of arbitrary-precision arithmetic without doing so, arbitrary-precision arithmetic should then generally be the default choice.
This assumes that you care about making relatively strong guarantees about the correctness of your programs. If for some reason you don't, then sure, use ints and whatnot for everything. If you do, though, I suspect you'll find that it's easier to track down a performance bottleneck caused by using bignums than an obscure bug triggered by GCC applying an inappropriate optimization based on overflow analysis.
