Contrary to my guess in the comments, the answer is no: there does exist a model structure on globular sets (reflexive or otherwise) which is Quillen equivalent to the Kan-Quillen model structure on spaces.
To see this, note that if $\mathcal A$ is a reflective subcategory of $\mathcal B$, and if $\mathcal A$ has a model structure, then the model structure transfers to $\mathcal B$, and the resulting adjunction is a Quillen equivalence.
Now, as mentioned in the comments, the category $Gph$ of graphs (reflexive or otherwise) is a reflective subcategory of $Glob$. Moreover, the category $Pos$ of posets is a reflective subcategory of $Gph$. Thus $Pos$ is reflective subcategory of $Glob$. So it will suffice to find a model structure on $Pos$ which is Quillen equivalent to topological spaces. This is proven by Raptis, by transferring the Thomason model structure on $Cat$.