Covers are explicitly allowed, even today. There's a standard royalty that applies, but no permission, or approval process or anything.
Michael Jackson made billions off the Beatles catalog, for example. The songwriting copyrights, not the recordings.
"Since the Copyright Act of 1909, United States musicians have had the right to record a version of someone else's previously recorded and released tune, whether it's music alone or music with lyrics.[7] A license can be negotiated between representatives of the interpreting artist and the copyright holder, or recording published tunes can fall under a mechanical license whereby the recording artist pays a standard royalty to the original author/copyright holder through an organization such as the Harry Fox Agency, and is safe under copyright law even if they do not have any permission from the original author. Other agents who can facilitate clearance include Limelight, the online mechanical licensing utility powered by RightsFlow. The U.S. Congress introduced the mechanical license to head off an attempt by the Aeolian Company to monopolize the piano roll market.[8]"
[1] https://en.wikipedia.org/wiki/Compulsory_license#United_Stat...
Any artists who haven't allowed their recordings to make it to electronic distribution are also drowned in poor covers online. AC/DC was a great example of this problem until they finally gave in and released their stuff digitally. There's still a ton of AC/DC cover recordings on iTunes.
But this is really beside the original point. The point is that copyright was intended to promote the progress of science/arts/knowledge and there are a lot of things right now that don't fall into fair use and hinders progress.
Here are some other examples. Many years ago when the Internet was first going mainstream, fan websites of TVs and movies were getting cease and desist letters because they had pictures of the actors in their roles on their websites. Technically this is still a violation, but most companies now at least have the sense to turn a blind eye. You'll see even on Wikipedia, there are a lot of reservations of posting a screenshot of a tv show, movie, or video game because of this.
Another random example is fan fiction. This seems like it would be a good candidate for rules to relax over time. At first, fan fiction may more restricted because the original authors may in fact want to write their own sequels. But over the course of time the work enters the public stream of consciousness and progress via things like fan fiction should be allowed instead of hunted down and destroyed.
Another example comes from an actual case law I can't remember the details from, but a book essentially about 'The most important things you should know when dating a Trekkie'. It made references to episodes in Star Trek so they got sued. Because it didn't directly parody Star Trek, it didn't fall under fair use. If I recall, the book lost the case. Again, since Star Trek has been deeply absorbed into the culture, it makes sense that copyright protection needs to weaken over time to allow more ideas/expression instead of hinder them. (A related topic might be the subtle distinction between parody and satire which is also currently a problem in current copyright law.)