A strong correlation between cooperation and membership in international institutions is not enough to establish that international institutions cause cooperation. If we’re to claim that institutions matter, we need to at least identify mechanisms by which institutions might promote cooperation among actors who would otherwise be disinclined to cooperate with one another. The mere fact that such mechanisms can be articulated does not itself tell us whether the correlation is causal, but it lends a certain measure of plausible to causal interpretations that would otherwise be lacking.
If you grant that international institutions matter at the margins, you’ve already conceded that they make a big difference to the overall level of cooperation we can expect to observe in the international system. See this post over at my personal blog for an explanation.