Introduced in [BB], simplicially stable spaces are alternative compactifications of $\mathcal{M}_{g,n}$ generalizing Hassett's moduli spaces of weighted stable curves. We give presentations of the Chow rings of these spaces in genus $0$ using techniques developed by the author in [New25]. When considering the special case of $\overline{\mathcal{M}}_{0,n}$, this gives a new proof of Keel's presentation of $\operatorname{CH}(\overline{\mathcal{M}}_{0,n})$.