From the comments in this thread the Twitter copy was not the original creator of the video. The original creator posted it on YouTube but has since made the video private. And they DMCAd the person that reposted it on Twitter.
I imagine they probably started getting a lot of negative attention.
Anyone is physically capable of sending a DMCA takedown.
However it is supposed to be from the copyright owner [17 USC 512(c)(3)(A)(i)] and if it isn't, it can be ignored [17 USC 512(c)(3)(B)] and makes the person who does it liable for damages [17 USC 512(f)(1)]. Although that's probably pretty hard to use in practice.
Your YouTube support link isn't about the DMCA, and in fact is an attempt at trying to get people to use their system instead of the DMCA.
YouTube's internal copyright takedown system is just that: Internal. DMCA is a legal avenue that you can use against YouTube and their users, but it wouldn't be via their internal copyright take down system.
It is in YouTube's best interests for you NOT to use the DMCA as there are avenues where it can leave YouTube themselves civilly liable (OCILLA[0]), so they work hard to funnel people into their internal copyright system that doesn't expose YT.
OP’s point is that anyone can lie in the DMCA takedown form and say they own the content. It’s illegal but I’m not aware of it ever coming back to the perpetrator in the form of jail time/lawsuit.
