Yes, CT "hardnes," "shape" and "condition" all affect draw - not to mention all other forms of english as well.
But those three factors combined - only contribute maybe 30% - to the AMOUNT of spin induced. Stroke speed, location of the CT in relation to the CB dead-center and other nuances control the remaining 70%!
I urge you to briefly focus on the CT on your stick. In theory, the CT never touches the CB. The abrasive material ON the surface of the CT - does!
THAT material is chalk. The "condition" of the CT determines the shape and condition of the chalk applied thereto. We shape the CT, we rough it up with sandpaper, burnishing, a tool with tiny needs and shaping abrasives to get the CT to a shape we desire, then cove it with chalk to make the CT "sticky."
Among the biggest regrets of someone making a multi-ball "run" is a miscue. Miscues are mostly caused by bad aiming, but not that infrequently, by striking a dry spot on the edge of the CT - with little or no chalk on it.
To illustrate, chalk is like sand. If you're on a sheet of ice, trying to get traction, a little sand is great. Too much sand and the tires will spin on just the deep sand alone, e.g., like driving a car on the beach. Too little sand and the slipping is inconsistent.
Chalking up is similar. Too much in the wrong place - is as bad as none - anywhere.
If you were to do a "forum search," There's a ton of threads involving tip shape and chalking up. All better than what I could write.
Good luck and keep us posted as to what yo find.