The basic issue is that not every function that we would like to describe between $Delta$-complexes can be realized by a natural transformation between functors. The lack of degeneracy maps means that no map $X to Y$ of $Delta$-complexes that sends any simplex down to a degenerate simplex can be realized by a natural transformation of functors. For example, if $X$ is a $Delta$-complex interval realizing $[0,1]$ and $Y$ is a $Delta$-complex realizing $[0,1]^2$, then there is no natural transformation of functors realizing the projection maps $p_i:[0,1]^2 to [0,1]$.
As a consequence, the category of $Delta$-complexes does not have enough immediately-available maps between objects to construct the kinds of colimit diagrams one would like to realize.
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