Somebody has to pay anyway
This is one of these cases where I can empathise with both sides of the dispute, up to a point.
I do publish a fair bit in scientific journals, and it is clear that Elsevier is currently interested purely in maximizing their profits (this has not always been the case). For this reason, I try to avoid publishing in and refereeing for Elsevier journals. I do not boycott them altogether (this would be professionally untenable), but if an alternative exists I try to choose that. Nonetheless, I would prefer if they continued to exist, preferably in somewhat diminished and humbler capacity.
The reason is that scientific publishing is not about simply sticking your work online, and hoping somebody will read it. There is a number of additional conditions, which make all the difference, namely:
1. Quality assurance. There is nothing your average scientist likes to grumble about more than peer review and the incompetence of the referees of her or his latest paper. Nonetheless, most scientists will also agree that peer review is an essential step of providing at least some assurance the work is not totally bogus. Having a formal (ie peer-reviewed and published!) comment and retraction system also helps.
2. Attribution and authentication. With a journal publication, I have a good degree of assurance that the work is done by the people appearing on the byline. For publications which are not too old, I also have a way of contacting the actual authors of the work. Of course, the system is not infallible - but failures are hoaxes are rare (which is exactly why nearly every case becomes news).
3. Immutability and persistence. With a journal publication, I have an expectation that the resource will not change between now and tomorrow. When a correction is required, then an erratum will be published, and will also become part of the permanent record. Furthermore, I also have an expectation that the article will still exist five, ten, or hundreed years from now. (It is not uncommon for me to have to refer to scientific articles published in in the late 19-th and early 20-th century; on one occation, I also found an article from early in 18-th century to be what I needed).
4. Indexing and cross-referencing. With a journal publication I expect to be able to find the articles I need in professionally-maintaioned subject databases. Sometimes these will ne online; somethimes not. Google and the ilk may be useful for some of the more superficial quieries, but sooner or later real understanding and knowledge is required.
A lot of these tasks can be done by the scientists themselves "for free", as a contribution to the community (I certainly do a fair share of refereeing myself). However, in the end, not everything can be done by volunteers, and somebody has to pay for the administrative expenses, the purchase and upkeep of the hardware, communications costs, backup and archival, and so on.
If we want to do a good, reliable job out of it we'll likely need to pay a lot: few of top-level experts in any field are willing or able to work entirely for free.
We can choose to pay these costs upfront as the society, in recognition of the importance of the scientific information systems for our continuing prosperity (or perhaps even survival). Or we can choose to pay piecemeal, through a multitude of commercial publishers. I do not really care which it is - but somebody has to pay in the end. And I do not see how scihub would be able to pick up the bill, or fill the roles played by the scientific publishers - commercial or otherwise.