The following result of Howe [https://zbmath.org/0844.20027][Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond] answers this completely:
Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. Then a non-trivial irreducible $\mathfrak{g}$-module $V(\lambda)$ has one-dimensional weight spaces if and only if
- $\lambda$ is minuscule,
- $\lambda$ is quasi-minuscule and $\mathfrak{g}$ has only one short simple root,
- $\mathfrak{g}=C_{3}=\mathfrak{sp}_{6}$ and $\lambda=\omega_{3}$, or
- $\mathfrak{g}=A_{l}=\mathfrak{sl}_{l+1}$ and $\lambda=m\omega_{1}$ or $\lambda=m\omega_{l}$ for some $m\in \mathbb{N}$.
I was able to find this via a paper of Stembridge [https://zbmath.org/1060.17001][Multiplicity-free products and restrictions of Weyl characters].
As I needed exactly this property in a paper of my own, you will find definitions of minuscule and quasi-minuscule as well as references from which I drew a table of these on pages 20-21 of it (arXiv:0409359, https://arxiv.org/abs/math/0409359On Lie induction and the exceptional series). They are as follows, taken from [https://zbmath.org/0957.17006][Visual basic representations: An atlas] by Plotkin--Semenov--VavilovPlotkin–Semenov–Vavilov:
Non-zero minuscule weights:
$\begin{array}{ll} A_{l} & \omega_{i},\ 1\leq i \leq l \\ B_{l} & \omega_{l} \\ C_{l} & \omega_{1} \\ D_{l} & \omega_{1},\ \omega_{l-1},\ \omega_{l} \\ E_{6} & \omega_{1},\ \omega_{6} \\ E_{7} & \omega_{7} \end{array} $
Quasi-minuscule weights:
$ \begin{array}{ll} \begin{array}[t]{ll} A_{l} & [1,0,0,\ldots,0,1]\ (\mbox{adjoint}) \\ B_{l} & \omega_{1} \\ C_{l} & \omega_{2} \\ D_{l} & \omega_{2} \\ E_{6} & \omega_{2} \\ E_{7} & \omega_{1} \\ E_{8} & \omega_{8} \\ F_{4} & \omega_{4} \\ G_{2} & \omega_{1} \end{array} \end{array} $
