There are counterexamples for every integer $k\ge3$.

In fact, if $2\le k\lt\omega$, there is a $k$-uniform hypergraph $H=(V,E)$ such that $|V|=\aleph_0$, $\{e_1,e_2\}\in\binom E2\implies|e_1\cap e_2|\le1$, and $H$ has chromatic number $\chi(H)=\aleph_0$.

Namely, let $V=\binom\omega{k-1}$ and $E=\{\binom X{k-1}:X\in\binom\omega k\}$.

It follows from Ramsey's theorem that $\chi(H)\gt n$ for each $n\lt\omega$; the other properties are obvious.