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The credit to an institution does not come from visits to its website, but from downloads and citations of its research output.

The purpose of Open Access is to maximize the downloads and citations of research output.

Hence it is definitely worth the small cost for an institution to host a repository to make its research Open Access.

And to make sure the repository is filled with all the institution's research output, the institution (as well as the funders of the research) should mandate that all of their research output is deposited in the institution's repository.

As for central repositories like Arxiv: They may well require local downloads to cite as a justification for the expense of their upkeep. My own conclusion is that central repositories are obsolescent and not worth the high cost of hosting the research output of institutions worldwide centrally. It makes infinitely more sense for institutions to host their own research output, thereby distributing the costs across institutions, and then leave the rest to central harvesting. 

(Ditto for distributing the "costs" -- i.e., keystrokes -- of deposit: the mandates should be that researchers do their own self-archiving, rather than institutions having to pay central staff to do it for them: it's just a few extra keystrokes per paper, and soon becomes a habit and a normal part of the researcher's workflow.)

And don't worry about where the downloads are occurring: The institution gets the visibility and impact from the fact that it is that institution's research output that is being downloaded (and cited). 

And, as I suggested earlier, there will be better and better tools for collecting pooling distributed OA download counts, as more and more research is being made OA: The way to accelerate that it is not to try to prevent harvesting, but to mandate deposit.