So what I'm asking is how does a quantum computer solve quantum chemistry problems? To make it concrete, say I want to calculate the band structure of a crystal using a quantum computer? What do I do? What kind of algorithms can the QC run that solves this problem faster/better than, say, a DFT code?

I don't think it's a given that just because I want to simulate a quantum mechanical system in some way, that a quantum computer is a useful tool. Or it might be, I don't know. I just don't think it's enough justification that there happens to be the word "quantum" in both the problem description and in the name of the computer.