I think another obstacle for midlevel players is... they read about the theory of tangent lines
and such, but they don't hit the cue ball consistently enough to make the the cueball path
match the diagram in a book. Like the accidentally allow it to pick up topspin, or draw a little,
or the ball picks up sidespin on the rails, etc. So they kind of lose faith in tangent lines
and stop thinking about the EXACT path of the cue ball. Later if you can show them how to make
a ball move exactly on that line, it becomes clear it's not just "on paper", it has practical use.
I couldn't agree with this more. Half the time when I coach I say "Just shoot a stop shot, it'll work out great" to weaker players on a cut shot, and when they actually shoot the stop shot correctly, go along the tangent line, and get shape, they'll invariably come up frowning and say "I guess I got shape, but I didn't make it stop." I'm thinking, what are they talking about, that was perfect! At that point I realize how little they understand about the tangent line-stop shot relationship.
These days, 9/10 times I help a beginning player, I try to explain the slide-stall-roll of the cue ball first. Amazing how many look at me like I'm talking voodoo science they've never heard before.