(I guess this is an elaboration on one of your comments in your last paragraph.)

One reason why the Zariski topology is "bad" is that the higher cohomologies ($H^i$ for $i > 0$) of a constant sheaf on an irreducible space (in particular irreducible varieties with the Zariski topology) are zero, because such a sheaf will be flasque (exercise!).

I guess this is one of the motivations for the etale topology and for etale cohomology. The etale cohomology of (certain) constant sheaves will be nontrivial, indeed, it will coincide with the singular cohomology of the associated analytic variety. This is the "Comparison Theorem", see for example Milne's book on etale cohomology for the details.