It's easy to show that one cannot have _two_ real characters $\chi, \chi'$ with $L(1,\chi)=L(1,\chi')=0$, as then $\zeta(s) L(s,\chi) L(s,\chi') L(s,\chi \chi')$ would have a  simple zero at s=1 while having non-negative coefficients, which is absurd.  This is of course a minor variant of the argument that rules out vanishing for a complex character (and its complex conjugate).  The intuition here is that the primes can almost conceivably conspire to be almost totally correlated (or more precisely, anti-correlated) with one character, but not with two; see  [this blog post of mine][2].

In some sense, the fact that $L(1,\chi) \neq 0$ with at most one exception is the best one can do using without introducing algebraic number theory methods (class number formula) or moving away from s=1 (in particular, using s=1/2 information); this is discussed in [this paper of Granville][1].

The fact that nobody knows how to make the constants in Siegel's theorem effective seems to me to be a significant upper bound as to how short or elegant a known proof of $L(1,\chi) \neq 0$ can be; at some point one must perform some maneuvre that is very expensive with regards to effective bounds (e.g. moving one's attention from s=1 to s=1/2, or bounding the class number from below by 1).


  [1]: http://www.ams.org/mathscinet-getitem?mr=1220462
  [2]: http://terrytao.wordpress.com/2009/09/24/the-prime-number-theorem-in-arithmetic-progressions-and-dueling-conspiracies/