The cup will work! The tape you used was not laid down smoothly enough for the cup to grab and/or the tape was textured like duck tape.
Keep in mind you also need to apply heat to the display so the adhesive softens. I would apply the tape mostly in the middle leaving a small margin around the edges so you can get to the area where the display is adhered with a heat gun or iOpener without heating the tape directly as well as make sure I don’t create to many wrinkles in the tape for the better suction. Work on one side at a time and use the picks and/or cards to help hold the glass from re-adhering going in a clockwise direction.
To help in getting the cup to adhere you can apply a bit of petroleum jelly to the cup to help improve the seal. You’ll want to fully clean the cup afterwards as applying it directly to glass will leave a good mark which will be hard to clean off.
Don’t forget to wear safety glasses!!