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.

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.

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).

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.

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.

The fact that nobody knows how to make the constants in Siegel's theorem (the one on Siegel zeroes) 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).