The wand has limited charges, and the spell is concentration
I'm assuming that you're using the Wand of Magic Detection in the DMG. If so, that wand only regains 1d3 expended charges per day. That means that at most, they will be able to use it 3 times a day, putting a hard cap on the number of times they can use the spell.
You point out that the duration is 10 minutes, but it's also a concentration spell. If the party gets into a fight, the wizard has to drop the spell in order to cast any other concentration spell, and they might lose concentration as a result of taking damage anyway.
Detect Magic is nonspecific and can be tricked
Detect magic only tells you that magic exists, and the school it has. It doesn't tell you anything more specific than that.
You can take advantage of those limitations to build interesting encounters centered around false positives, where detect magic will mislead the PCs.
For example, you could have two paths in a room. On one path, there is a 10'x10' patch of illusion magic, and on the other, there is nothing. If the PCs are relying only on their wand, they might think that there is a trap under the illusion magic, but the illusion magic is just an illusory cover of dust, and the non-magic path contains a mundane trapdoor.
As for your hag scenario, the hag could have cast an illusion on both the NPC and herself, which would make them equal in Detect Magic's eyes.
Indeed, this kind of trickery is supported by the rules: Nystul's Magic Aura is a spell that lets you change how magic is detected:
You change the way the target appears to spells and magical effects, such as detect magic, that detect magical auras. You can make a nonmagical object appear magical, a magical object appear nonmagical, or change the object’s magical aura so that it appears to belong to a specific school of magic that you choose.
As with any trick, you probably don't want to overuse this spell, but a few good applications of Nystul's Magic Aura will teach your party not to rely too heavily on Detect Magic.
Is it really "abuse"?
If I were the DM, I'd think that the PC's use of their wand to pierce the illusion was an acceptable and clever solution to the puzzle. Remember that the PCs are (generally) intended to solve the problems they face, and they are expending limited resources to do so. Speaking from experience both as a DM and a player, it might feel anticlimatic as a DM to have your encounter cheesed like that, but your players might feel quite good about solving the problem so definitively.