This requires some topos theory I'm afraid, but gives, I think, a satisfactory explanation.

The category $\mathrm{Set}^{\Delta^{\mathrm{op}}}$ $\textit{classifies intervals}$: the simplicial set $\Delta[1]$ is linearly ordered, with smallest and largest elements $0,1:\Delta[0]\to\Delta[1]$, and for any topos $\mathscr E$ and any linearly ordered object $I\in\mathscr E$ with distinguished smallest and largest elements $\bot,\top:1\to I$ there is a unique up to isomorphism $f_I:\mathscr E\to\mathrm{Set}^{\Delta^{\mathrm{op}}}$ with $f_I^*(\Delta[1])=I$. In particular, one can form various $\mathscr E$ containing one or other category of spaces, and for the corresponding interval $I$ there, $f_I^*(S)$ is the geometric realization of the simplicial set $S$, while for a space $X$, ${f_I}_*(X)$ is the singular simplicial set of $X$.

Thus simplicial sets are determined (up to equivalence of categories) by the fact that they classify all possible notions of a continuous path, hence of a homotopy.

Reference - e. g. in ncatlab; first occurrence of this I've seen is in Johnstone's first topos theory book, where it is attributed to Joyal.

This requires some topos theory I'm afraid, but gives, I think, a satisfactory explanation.

The category $\mathrm{Set}^{\Delta^{\mathrm{op}}}$ classifies intervals: the simplicial set $\Delta[1]$ is linearly ordered, with smallest and largest elements $0,1:\Delta[0]\to\Delta[1]$, and for any topos $\mathscr E$ and any linearly ordered object $I\in\mathscr E$ with distinguished smallest and largest elements $\bot,\top:1\to I$ there is a unique up to isomorphism $f_I:\mathscr E\to\mathrm{Set}^{\Delta^{\mathrm{op}}}$ with $f_I^*(\Delta[1])=I$. In particular, one can form various $\mathscr E$ containing one or other category of spaces, and for the corresponding interval $I$ there, $f_I^*(S)$ is the geometric realization of the simplicial set $S$, while for a space $X$, ${f_I}_*(X)$ is the singular simplicial set of $X$.

Thus simplicial sets are determined (up to equivalence of categories) by the fact that they classify all possible notions of a continuous path, hence of a homotopy.

Reference - e. g. in ncatlab; first occurrence of this I've seen is in Johnstone's first topos theory book, where it is attributed to Joyal.