They don't own a theme or video format, they own a trademark, which falls under an entirely different set of rules from copyrights and patents. Any videos being taken down on YouTube because the Fine Bros filed a copyright claim is entirely the fault of YouTube for having a shit copyright system. If those videos had been published on any other website, the chances of them having been brought down is slim. Furthermore, any copyright claim against other reaction based videos in a court of law would have ended poorly for the Fine Bros.
A trademark is an identifier. For example, Coca Cola and Pepsi both make a drink called cola. But Coca Cola can't use any of the names, logos, slogans, etc that Pepsi uses as part of its business nor can Pepsi use Coca Cola's names, logos, slogans, etc. If companies were allowed to use each others identifiers, then consumers would quickly become confused as to which company they were buying from.
Likewise, if you created a show on YouTube, for monetary purposes, which depicted children reacting to various cultural memes and entitled the show "Kids React", then the Fine Bros would ABSOLUTELY have the right to order you to change the name of the show. If, however, you entitled the show "Children Respond", and used none of the logos, slogans, artwork, etc that the Fine Bros use in "Kids React", then the Fine Bros wouldn't be able to do shit to stop you (legally, as stated earlier, YouTube's copyright system is shit).
Furthermore, trademark law is not explicitly in the US Constitution. In 1879, the Supreme Court ruled in the Trade Mark Cases that the Copyright Clause in no way gives Congress the power to regulate trademarks, instead, Congress enforces trademarks via the Commerce Clause which means the First Amendment applies in full. On top of that, there are provisions in US law allowing for fair use of trademarks. Because trademarks are more limited than copyrights, the standards for fair use much lower. The use of someone else's trademark to accurately describe your product or to refer to a product/service which is trademarked are both considered fair use.
In summary, while we can all agree that the Fine Bros were behaving in an absurd way, YouTube's copyright system is the real villain here as it is easily used by some content producers to shut down competition. Chances are good that none of the Fine Bros copyright claims would have held up in court. Furthermore, the fact that the Fine Bros own a trademark on the names of their shows is no more evil than Canonical Ltd owning a trademark on the African word "ubuntu" (Registration number is 4578010) or Microsoft having a trademark on the word "windows" (Registration number is 4407849).
If you're still concerned about possible First Amendment issues, I recommend you read Rogers v. Grimaldi. You only have to read the background and section I under discussion. TL;DR The court did an amazing job balancing the First Amendment and the Lanham Act (trademark law).