Q:

Where is my reasoning wrong? 

We know that components of locally connected sets are open.  Now consider the space
$$X=\bigcup\limits_{q \in \mathbb{Q}}^{} \{(x,y) \in \mathbb{R}^2; x^2+y^2=q^2 \}-\{(0,0) \}$$
Now this seems locally connected, since every point belongs to a circle and you can just take smaller and smaller arcs around that point.  However the components are the circles $x^2+y^2=q^2$ which aren't open. 
Edit: $q \in \mathbb{Q}^{+}$ not $q \in \mathbb{Q}$

A:

It’s not locally connected at all: any point on the circle has basic neighbourhoods that are intersection of open balls in the plane with $X$, and these will contain many points from other rational-radius circles.  So your arcs aren’t neighbourhoods at all.  
You treat the circles like they’re a topological sum, not as a subspace of the plane.