You bring up a good point. If a hole is bottomless, it leaves Earth and goes into space, where it automatically becomes infinite.
Oh. I hear “Also Sprach Zarathustra” playing in the background.....
Problems with dropping a line with a weight too. Upon reaching the center of the earth, it would stop “falling”. Although at that depth you wouldn’t be able to feel it if it did stop.
If the Indian is telling the truth with his lifesavers trick (a good idea!) - if that is all that he had on the line, after 300’ it very well could have hit the dry bottom of the hole and he would not have noticed it hit bottom. (The weight of the line being more than the lifesavers at that point.)