Assume a distribution $X$, and we know the idf $F^{-1}$ of $X$. Let $U \sim U(0,1)$.
Why is drawing an element from X according to $F^{-1}(u) = Z$ considered to be more random than just drawing an element from $X$? Edit: It's not.
My guess would be that in an unevenly distributed set, there are areas that are more concentrated, and thus have a higher chance of the item being picked from there. Violating the uniform distribution property.
By sampling from a uniformly distributed set $U$, you somehow bypass that. I get that for every sample from $U : u$, you get a corresponding minimal $x$ from $X$ where $P(X < x) > u$. But what I don't get is how we bypass the chance of drawing more elements from more concentrated areas.