As a minor objection to your analysis, some of the papers they distribute are by employees of the US government. There is no copyright for those papers, therefore Elsevier could not get anything signed over to time.
Elsevier and all other journals charge the same price for access to these papers as they do for papers where the publisher controls the copyright. To a first approximation, that means the value of the copyright is not significant.
It's a minor objection because there are few papers which are in the public domain. The publishers use contracts to prevent or limit redistribution.
(I have wondered if a SciHub variant with only papers which are out of copyright or not under copyright would be viable, even on a US server. I invite someone else with money and lawyers to try it out.)
Anyway, to get back to your AWS server scenerio, the parallelism doesn't work that way. In the print publication era, multiple organizations functioned as an archive for the material, including the institution library, various university libraries, and state libraries. I have gone to the local(ish) chemistry library and read publications from the 1960s without paying extra costs to read the article or make copies. I can just walk in as a member of the public and do that.
However, it's more cost effective to centralize things at Elsevier (that savings to the library get passed along as profit to the publisher), Elsevier moved towards an electronic-only publication system mediated through user agreements instead of copyright, so the library cannot act as an archive, and that user agreement must extend to all users, including members of the public. (I have signed a paper to that effect, which lets me as a member of the public to access online publications but without the right to print them out or make copies.)
In your AWS analogy, Elsevier/Amazon once distributed their software to anyone, and there were dozens of places willing to provide $5K/year to host those services for users of the local cloud, including anonymous hackers. Elsevier/Amazon decided to centralize, by only hosting new services on their machines rather than distribute the software. They also decided to require verified logins. So now our anonymous hacker is forced to either give up doing science, pay potentially $1,000s for legit access, or use subterfuge to do what was once inexpensive or even free, as part of a library's mission to support the public.
But in real life AWS never distributed all of its code, so that others could provide AWS services, which is why your analogy doesn't really work as a metaphor for this issue.