Waving a wand and wiggling the fingers while magic happens is theatrics. In Faust, Goethe has the Devil make fun of a witch for being too precious and ceremonious with her magic, so you are in good company cutting out the silly hand gestures.
Addressing copyright fears, wands are "stock items" in stories with magicians and wizards. No one can claim to own them.
However, the Potter stories feature magic wands as a rightrite of passage, as several MacGuffins, and as a symbol for the absolute power which Harry rejects. Wands take up quite a bit of story real estate in a very prominent commercial franchise, and no living person has escaped exposure.
I love the way that wands are like gunfire and how you flick and attack and it's instant. No weird hand movements or long unwieldy staffs or canes to use.
J.K. Rowling's current spin off series – which appears to be sinking – has reduced the wand's narrative contribution to "pew pew pew". You have your own wand idea, so I'd encourage you to go deeper, not reductive.
I suggest if you want your magic to look like a Tesla™ plasma zapgun, give it some logical constraints that fit. If it shoots lightning then it is depleting ions and starting fires. There may be limits to how quickly you can "pew", how far it shoots, or the dangers of standing in water or near a conductor. Constraints are usually better narrative tools than superpowers.