Well, the problem is that all this "vigilante[0] enforcement" was there to enable online services to work at all; and it has been further extended by private business agreements to give copyright owners powers above what the law requires.
Any tort in law also has secondary liability attached to it, and it's generally accepted caselaw that online service providers get secondary liability if someone uses them to infringe. Congress decided that said providers should be able to disclaim that liability if they participate in a notice-and-takedown process; and the EU has gone further and replaced that with a very context-sensitive "best practices" approach that almost certainly will be interpreted as "video sites must have an equivalent to YouTube Content ID".
From an old media perspective, this is crazy. If CBS licenses a TV show that infringes upon a third-party copyright, they cannot disclaim liability by saying it's the licensee's problem. If they could, then you could construct a bunch of LLCs to separate the infringement from the liability. And this is effectively what online services have done: YouTube, for example, gets to automatically[1] recommend you infringing content, and those who have been infringed only have the legal recourse of sending takedown notices.
Getting rid of the takedown system would mean that copyright owners would have to go back to suing individual users, which is both expensive and, IMO, wrong. YouTube gets to own online video and infringe copyright with impunity, because they passed the buck onto individuals that largely cannot afford to pay damages on infringement. If you're a legitimate[2] user of copyright, you can't economically get the infringement to stop or go underground. Furthermore, if you are a copyright troll, this does nothing to stop you; you will still be able to sue individuals and coerce them into settling baseless claims.
The underlying problem is that online platforms want to half-ass copyright. You can decide whether this is because they understand the draw that infringing content provides to them, or if proper enforcement interferes with the FAANG business model of "own the platform and take 30%". Either way, it's a compliance cost. We don't want people just suing individuals, and we don't want people suing online services to the point where any amount of third-party content is too legally perilous to touch. The answer was supposed to be notice-and-takedown, which is designed basically as a way for online service providers to push papers around between copyright owners and users without having to resort to an expensive lawsuit.
However, nobody likes this. Online service providers that try to do things "by the book" wind up with all sorts of legal pitfalls for both themselves and their users. So even the current notice-and-takedown regime in the US has been subverted by private agreement. For example, DMCA 512 requires copyright owners to actively monitor infringements of your work; but YouTube Content ID instead provides a turn-key solution to find infringing work on that platform. It also lets you just monetize the infringements instead of taking them down, which is also a sort of backdoor license for Google. I suspect some sort of similar arrangement is involved with Google Drive, where you can't share files that trip whatever content ID system is involved here.
So even "killing vigilante enforcement" would not be good enough, because Google has already found it lucrative to privilege copyright owners over regular users. You need to specifically regulate these takedown alternatives rather than just getting rid of takedowns.
[0] Technically speaking, a DMCA takedown notice from someone who does not own the copyright to the work in question is legally deficient and can be safely ignored by Google. However, most online service providers do not do a good job of checking.
[1] As per Mavrix v. LiveJournal, the process has to be automatic. Human curation takes you out of your DMCA 512 safe harbor. This is effectively a "willful blindness" requirement, IMO, which is also bad.
[2] As in, you're interested in preventing sharing of newly-published works for a reasonable time frame, rather than keeping an ironclad grip on Steamboat Willie or Super Mario Bros forever.