> the gaps are reduced to zero and the curve crosses through each and every point within its build envelope
I think that's oversimplifying an important point. If you build a Hilbert curve in a 1x1 square, the vertices of the curve always have rational coordinates. So all points on its line segments must always have at least one rational coordinate. There's no way it can cross through every point in a square region of R^2.
A better way to say it might be "the gaps are reduced towards zero and the curve will pass arbitrarily close by every point in its envelope". That still explains why its Minkowski dimension must be 2.
Most simply, after "infinitely many" steps, the "arbitrarily close" in your mental model becomes "infinitely close", and in real numbers, "infinitely close" is the same as "equal to", because you don't have infinitesimals.
It's the same reason why we can construct irrational or transcendental numbers as a limit of an infinite series of rational numbers.