Data summarization techniques, such as coresets, are essential for scaling clustering algorithms to massive datasets. We study uniform sampling, which is an extremely simple and ultra-efficient method to construct coresets, in the context of weak, stable, and strong coresets for the $k$-means problem in Hilbert space. We begin with the fundamental case $k=1$, and establish matching upper and lower bounds on the sample complexity of stable and weak coresets, finding that the two notions require different sample complexities. As a consequence, we obtain tight bounds for stable $1$-median coresets in $\ell_1$ and $\ell_2$ (and thus also Kendall--tau and Jaccard metrics), removing the dimension dependence of Carmel and Krauthgamer (ICLR 2026). Our analysis is substantially simpler than prior work and reduces to bounding the deviation of the sample mean. For the general $k$-means problem on $ρ$-separated instances, a well-studied structural assumption on the input, we show that uniform sampling yields a stable coreset for all $O(1)$-approximate solutions, and a strong coreset if the instance additionally has bounded kurtosis. In all these results, the sample complexity is independent of the dimension and the instance size.

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