Giving him mage hand would be an awful idea, since AT is supposed to get it for free.
No it doesnt get mage hand for free, you get mage hand buffs.
According to 5e rules, Arcane Trickster does get mage hand for free:
Cantrips. You learn three cantrips: mage hand and two other cantrips of your choice from the wizard spell list. You learn another wizard cantrip of your choice at 10th level.