Because on Harry Potter, wands don't really do anything extra besides channel magic. You could argue that it's because it's a whole magic system etc. But remember that in all seriousness, Harry Potter works like D&D without spell slots or leveled magic. Spells need Somatic components which are replaced by wands, Verbal which is the utterance of the spells, and in the case of becoming an Animagi has Components to use magic.

It is a very good story, although it is quite confusing at first. I think my confusion is highly attributed to whether or not the MC is a game character from a D&D game, which makes sense as to why he wouldn't have memories of his parents and can "metagame" calling jargon from D&D, but the point that he can't use wands seems pretty stupid to me. He does not need spellcasting focus as a wizard, but that does not stop a wizard from gaining a spellcasting focus. A wand in Harry Potter could be counted as a magical item and if he does have other items that he's attuned to then systematically sure. But, spellcasting foci doesn't need attunement to functionally use, so why can't he use wands from Harry Potter.

That's fair. Given how unforgiving the Spell Slot system is in D&D it doesn't make sense for it to exist in a magical universe where generally magic could be used endlessly.

so, a little thing of nerdism, Invisibility would have been broken when he casted Dancing Lights. Albeit, Dancing Lights is a cantrip that does not use a spell slot, it is still considered a spell. The text on Invisibility says that the spell is broken when the target attacks or a spell is cast.