The other day in my string theory class, I asked the professor why we wanted to quantize gravity, in the sense that we want to treat the metric on space-time as a quantum field, as opposed to, for example, just leaving the metric alone, and doing quantum field theory in curved space-time. Having never studied it, it's not obvious why, for example, the Standard Model modified to work on a space-time with non-trivial metric wouldn't work.
The professor replied in a way that suggested that, once upon a time, this was actually a controversial point in the physics community, and there was a debate as to whether one should head in the direction of quantizing the metric or not. Now, he said, the general consensus is that quantizing the metric is the right way to go, but admitted he didn't have time to go into any of the reasons that suggest this is the route to take.
And so I turn here. What are the reasons for believing that in order to obtain a complete and correct quantum theory of gravity, we must quantize the metric?
EDIT: I have since thought about this more, and I have come up with an extension to the original question. The answers already given have convinced me that we can't just leave the metric as it is in GR untouched, but at the same time, I'm not convinced we have to quantize the metric in the way that the other forces have been quantized. In some sense, gravity isn't a force like the other three are, and so to treat them all on the same footing seems a bit strange to me. For example, how do we know something like non-commutative geometry cannot be used to construct a quantum theory of gravity. Quantum field theory on curved non-commutative space-time? Is this also a dead end?
EDIT: At the suggestion of user markovchain, I have asked the previous edit as a separate question.
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