In a sense, this is a dangerous question because it looks at history and assumes that things had to be the way they were.  It assumes that the Roman system of government was better for ruling an empire simply because it did rule an empire.  It could well be that the Spartan system, in particular, would have been just as good and that it was something else that allowed Rome to rule an empire while Sparta could not.

Athens's government does seem to be much less conducive to ruling an empire.  The reason for this is that its government was a direct democracy.  Athens was really only able to have this sort of a system because it had such a small body of citizens.  A system of government that is suited only for a few thousand citizens will not translate well to an empire.

By contrast, Sparta's system seems like it would have been more "portable."  To some extent, it suffered from the problem of being democratic.  But it was more top-down than Athens's system was.  The idea of having kings who were advised and checked by the gerousia and the ephors does not seem all that different from Rome's system of a Senate with various types of executives.

Overall, then, Rome's system was able to rule an empire because it was more top-down than that of Athens.  Sparta's system might have worked for an empire, but we are not able to know that because Sparta was not militarily or economically capable of conquering an empire.