>Inasmuch as I'm very eager to reach a conclusion, maybe all this story with the glue is nonsense.
Ay ay yi! This view is far too ad-hoc and dominated by more or less physical intuition to make much sense of mathematically.
I'll clue you in though on the primary reason today's mathematicians still care for that lonely point, even though it has no length or breath. Without every last point being there on the line, you fail to achieve "closure" for some operations. What's that mean? Well, look it up, but it says things like there's an answer for every addition, multiplication, square-root operation, etc. The ascending numbers systems like natural numbers, integers, rational numbers, real numbers, complex numbers, achieve closure for ever more inclusive sets of operations.
But if you really want to learn math, you need to at least get grounded in the ways that mathematicians approach these sorts of problems, even if later you want to poke fun at them. Your sort of free-form thinking is fun, but its not really math.