In brief, because Austria was not just modern Austria: it was all the lands of the Habsburg Monarchy (encompassing modern Czech & Slovak republics, hungary, croatia, slovenia, parts of poland and romania, and more).

Among German nationalists, it was an open question as to whether a united germany should include Austria, with its large non-German population.

Quick side note: it was not a German Republic that was formed following the Franco-Prussian war. It was a German Empire, centred on Prussia with its King as German Emperor. It left for the most part in place the existing dynasts of the lesser German states.

Austria was a great power in its own right: the end of the Franco-Prussian war marks the end of a long period of Austro-Prussian "dualism", where Austria and Prussia were in competition for hegemony among the lesser German states.

The unification of the southern German states with the Prussian-dominated North German Federation was a blow to the power and prestige of Austria, but its Empire would last until the end of the first World War.