Good question. What the counting argument above proves is that the cardinality of all continuous curves has the cardinality of the reals, without regard to smoothness.
You are right that I should have been more careful about my use of Brownian motion, because I indeed wanted to use it to obtain a “specific” (of course the result is random!) continuous graph that is nowhere smooth, indeed is fractal, and is (probably!) not symmetrical. The fact that you can generate such paths via Brownian motion is not relevant; that was just supposed to be an example.
As for the cardinality of all Brownian (let’s say obtained via scaled random walks so they have to be continuous) paths, all nowhere differentiable paths, and similar subclasses of all continuous paths, the cardinality is less than or equal to the cardinality of all continuous paths, so it cannot be more than the cardinality of the reals, and in any such example it should be more or less obvious that it is no less.
One thing which I was going to mention: as soon as you allow fractal curves you have the possibility that the curve itself has positive area! For instance a square of area 1 could be split into two pieces, each of which has area 1/3, and the curve which forms their boundary eats up the last 1/3 ! A pathological bisection, to be sure, but without imposing any smoothness condition it can happen.