I'm not saying how it should be implemented, I'm saying how it is implemented by a number of popular VPN services. Take up the argument with them. There are VPN providers that do it right on at least some platforms, but unfortunately the way most document it it's very hard to tell without experimenting with the client to verify.
As far as I know, use of the term "kill switch" closely correlates with an untrustworthy implementation. Consider the case of Mullvad who handle this a lot better and also decline to call it a "kill switch" for that reason. And that's not to say that Mullvad is perfect, easy to find forum threads by people who had traffic leakage for various reasons. I wouldn't trust anything you didn't set up yourself.
As far as I know, use of the term "kill switch" closely correlates with an untrustworthy implementation. Consider the case of Mullvad who handle this a lot better and also decline to call it a "kill switch" for that reason. And that's not to say that Mullvad is perfect, easy to find forum threads by people who had traffic leakage for various reasons. I wouldn't trust anything you didn't set up yourself.