Why do we need a theory of quantum gravity

As you might know, Einsteins equations are
[tex]R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R = 8\pi T_{\mu\nu} \,\text{.}[/tex]
##T_{\mu\nu}## is the stress-energy tensor of the matter content of the theory (electrons, quarks, gauge fields, ...). The problem is that the matter is described by a quantum theory (the standard model) and thus ##T_{\mu\nu}## should really be an operator ##\hat T_{\mu\nu}##. But if the right hand side is an operator, the left hand side should be an operator as well for the equation to make sense. Thus we try to find a quantum theory that describes the gravity part of the eqation.

You might ask: "Why dont we use ##\left<\hat T_{\mu\nu}\right>## for the right hand side?" People actually do this and it's called quantum field theory on curved spacetimes. It is expected to be a limiting case of the a full quantum theory.