Most likely the contract with Elsevier prohibits this. In the fine print, "unlimited access" is probably defined as something like "what we, the publisher, consider normal/reasonable reading speeds by a human using our site/server." At my academic library, for example, it's common, and apparently contractually sanctioned, for a publisher to temporarily IP-block a computer that they deem to be engaging in "excessive" or software-mediated downloading.

Not sure about countries with more sensible policy but in the UK, as an academic, you have to publish regularly to not have your research grant applications declined. This means that there are a lot of papers out there that are published for publishing's sake. It's a game you have to play for your career as a postdoc. Depending on the field, it can be hard to tell whether a paper was written for publishing's sake or to actually present a useful result from the abstract alone.

Because of effectively mandatory publishing, mirroring is quite a big effort and, from a publisher's point of view, relatively easy to spot. It is probably these countermeasures which we are talking about here wrt LSU. With research being ever more focussed and specialised, there are fields where there may only be a couple of papers and you may be the only person in your department that is interested in that particular paper in that particular year.

Now, you could cache papers as they are downloaded, but there are definitely ways of making papers cache unfriendly and the journals have probably already done this, given they really don't have a lot else to do. As others have mentioned, it is probably also a breach of contract to cache or mirror the papers.

Because they have unlimited access?