So what if Harry or another muggle raised wizard for that matter saw a movie or read a book and figured magic people use wands but they get confused and just use a regular stick but because they are so powerful they just use wandless magic but think they aren’t because they are using their stick/wand.
I’ve read a few where Harry does try and pick up a stick and it doesn’t work because it’s, well a plain boring old stick. However this time it does work due to how powerful he is. This can also be used with OC’s in your story.
I also think it would be funny if a lot of kids who become friends do the same thing, like great minds think alike and a bunch of magical friends pick up random sticks and go to Hogwarts casting wandless magic because they don’t know better.