Why paste instead of push
Server-side scanning assumes a repository, a branch and a pipeline. That assumption holds for code that already lives somewhere your tooling can reach. It does not hold for a function pasted out of an assistant, a vendor's example file, a prompt fragment being evaluated, or a prototype that will never be committed in its current shape.
A client-side check closes that gap. The file is pasted into a page, the rules run on the text, and the answer arrives before you have decided whether the code is worth keeping.
It also removes a quiet friction. Nobody wants to add a dependency, request a token or open a pull request to find out whether a snippet is safe to run, so in practice nobody checks. Making the check free and instant is most of the value.