Continuous maps from $S^1 \to X$ equivalent conditions

Continuous maps from $S^1 \to X$ equivalent conditions

The following are equivalent for a topological space X according to a
problem in Hatcher.
$1$)Every continuous map $S^1 \to X$ is homotopic to a constant map.
$2$)Every continuous map $S^1 \to X$ extends to a continuous map $D^2 \to X$.
$3$)$\pi_1(X,x_0)=0$ for every $x_0 \in X$.
I am examining the proof of $2$) implies $3$).
Proof- Let $h:S^1 \to X$, then there is an extension $k:D^2 \to X$. If we
let $j$ be the inclusion $S^1 \to D^2$, then $h=k \circ j$ and $h_*=k_*
\circ j_*$. Since $\pi_1(D^2,s_0)=0$ for each $s_0 \in S^1$, then h* sends
each element of $\pi_1(S^1,s_0)$ to the identity in $\pi_1(X,h(s_0))$.
So I have two issues. First, this only shows $h_*$ is the trivial
homomorphism not $\pi_1(X,h(s_0))=0$. Second, even if I fix that then I
seem to have only proved it for $x_0$ in the image of $h$. So if someone
could help me with this I would be grateful.