Could you use Augury to detect a cursed item by asking
"Am I safe to attune this item?"
The GM would have to say either "weal and Woe" if the item is both beneficial and cursed or simply "woe" if there is only a curse.
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Sure, this seems like an entirely uncontroversial use of the spell.
But do be aware that a GM might have a deity value magic items differently than a PC would. You might get "woe", not because the item is cursed, but because it is considered sacrilegious from the deity's perspective. Or a holy item might get a "weal" result, despite a curse on it, if the deity thinks you're going to be able to purify the item, or if it thinks not having you attune it would be worse (e.g. somebody else would be more vulnerable).
There is also a mild opportunity cost to routinely casting Augury for questions like this. If you have some other event come up on the same day that you might want some divine guidance on, you may not get a truthful answer if you cast Augury again.