Breakout had a spinner (dial) which works on exactly the same principal as a mouse albeit only on the horizontal axis, so it has a notched wheel inside bisected by an optical sensor that reads pulses of light as the notches rotate and convert them into a signal.
The problem you have with the joystick is its not an optical positional device like a mouse or spinner, with a mouse you can move 5 inches left and stop and the paddle on screen will also stop moving, with a joystick you have to return to centre to cease movement as moving it halfway left will still be sending a signal to the game telling it to carry on moving left. TBH there's not really an ideal solution as the game is designed for progressive optical input and your joystick can't do that.
The only solution is to adjust the sensitivity until you reach a spot where the paddle onscreen accelerates and decelerates at an acceptable level dependent on how far you are moving the stick.
From what you're describing I'm presuming the input you have is either digital or pot based analogue, in either case you need an optical device to properly address the original hardware. You have to remember the NES game was programmed from the ground up to react properly to digital input but the arcade game wasn't.