There's a big mistake that most open access publications are making: they require readers to visit the publication web site to read articles. That is, anyone can read articles for free at the web site, and even copy them for their own use, but you aren't allowed to copy articles for republication on another site.
The obvious problem with this is that the operation of the web site becomes critical. If the web server goes down, then readers can't access the articles. If the publication forgets to renew its domain name, or some country decides to blacklist the domain name, then readers can't access the articles. If the publication goes out of business, or the hosting company goes out of business, or a disk fails and the publisher doesn't have a good backup system in place, then readers can't access the articles. The web site becomes a single point of failure.
A less-obvious problem is cost. All-electronic publications can be run cheaply, but not if you need to hire and manage a staff to develop and maintain a fancy web site. Web site expenses are part of the reason that some open access journals charge authors thousands of dollars to publish each article, a charge that is out of reach for many authors.
The purpose of academic publishing is to further the advancement of science by disseminating peer-reviewed research as quickly and as widely as possible. Republishing—replication—is clearly aligned with this purpose, and restricting republication is clearly at cross purposes.
Here's what we should do: move to a publication model that encourages replication and republication of the entire contents of open access journals. For example, libraries should be able to republish journals, and their patrons should be able to read articles through the libraries' web sites. Libraries—or anyone else—should be able to copy not only the articles but the table of contents of the journal, as well as other metadata. This provides multiple continuously-tested backups of the publication which can even survive the publication going out of business: once an article is published, it will always be available.
Journals and authors may wish to prevent some republication, e.g., commercial (for-profit) republication, or republication without attribution. This can be handled as a copyright and licensing issue. Preventing all republication, however, is a mistake.