If DMCA claims were expensive, companies would bother requesting takedown for the most profitable movies/albums/games only and they wouldn't go on autopilot, claiming whatever they think might infringe their copyrights. But then the garage bands and individual artists/musicians would be left out of the game completely, they couldn't use DMCA to take down their music.
On the other hand, how do you prove you are the author unless you have lots of money for lawyers? If you say "I composed this song at home, and published it under CC licensing" and MAFIAA say they represent the copyright holder, is it not clear who owns what?
If there was a digital notary service where you could digitally sign and timestamp a file, that might help a bit. But the downside is that people could take public domain works or someone's un-notarized works, claim ownership, have it stamped and they would begin to hold a stronger position pretending to be the author.
I think this is somehow similar to how indigenous people wondered how the white man could possibly think of owning forests, rivers, and land. They considered Earth unownable, not belonging to anyone but rather people belonging to the Earth. Similarly, natives to the digital world wonder how somebody could possibly think of owning bits, numbers, and copyable files.