As I said, the Riemann zeta function has its definition because it makes sense and the other doesn't. Once you have a solid definition of ζ(-1), you could declare that 1+2+3+...=ζ(-1) and then you might be tempted to reverse the sign. But the zeta function was around for a century before Riemann encouraged people to emphasize the values that don't make immediate sense.

You can do an awful lot just having it defined for real s>1. Euler used it to prove the infinitude of primes: ζ(1) is the harmonic series, thus infinite (or more precisely, an infinite limit as s approaches 1), but prime factorization expresses it as an product over primes, so there must be infinitely many primes to make it blow up. Moreover, this gives a better estimate of the density of the primes than Euclid's proof. Then Dirichlet used it and related functions to prove that there are infinitely many primes satisfying reasonable congruences. (Exercise: use Euler's technique to prove that there are infinitely many primes congruent to 1 mod 4 and infinitely many congruent to 3 mod 4.)