As a constructivist, I somewhat agree with your viewpoint, though my views are a bit more nuanced. (e.g. Dedekind's construction of the continuum as a collection of points is perhaps the primary source of corrupted thinking, which muddles even well-meaning thinkers into thinking that it is even possible to separate computable points from non-computable points. The real numbers form a /continuum/, and a continuum cannot be described as simply a collection of points.)
That said I believe your comments are off the mark in regards to the topic of the Weierstrass function. The Weierstrass function is completely well behaved from a constructive point of view[0]; it is uniformly continuous and everything. In particular we can (constructively) map (constructive) real numbers to (constructive) real numbers via this function.