Most independent CFI's will have access to airplanes through an FBO ect. Some (like myself) instruct using their own aircraft. How quickly you progress by using an independent CFI, mainly comes down to your drive and the Instructors availability. The training is done under faa part 61 rules vs part 141 that most pilot factories operate under (I'm partial to part 61 training as it offers more flexibility in tailoring a students training to their learning needs).

On the whole, you'll probably wind up saving a ton of money training with an independent flight instructor. My suggestion is to get your Private Pilot Certificate with an independent instructor to see if it's a good fit for you continuing forward.