The debate over voter ID isn't a rational debate about how to make elections more secure. It's just another battleground in which the Republican Party sees an opportunity to shift election outcomes slightly in their own favor. They're not interested in a rational, balanced solution like you're proposing, because it might end up shifting election outcomes in the "wrong" direction.
When you look at the list of election measures the Republican Party supports/opposes, it becomes quite clear what their goal is. They're against early voting, automatic voter registration, absentee voting (except for the military) and making election day a holiday, but they're in favor of voter ID and they engage in very heavy gerrymandering (leading to a divergence between national vote shares and numbers of seats won).
From where I sit it's a win. Administrative work will be easier, all cases of fraud will be harder, and arguments of nutjobs weakened. Not to mention the fact that people would trust the election process more, with all the added benefits.
Really, voter IDs seem like a stupid hill to die on. Especially if you are already dealing with low trust in political system and cultural divides.