I read the help page.

https://www.deadmansswitch.net/help/

How can I be sure you'll outlive me?

We don't have to outlive you! If the service shuts down while you're alive, we'll send you an email well in advance so you can switch services. That said, the service has been running successfully since 2007.

In the spirit of the OP, and finding an enduring solution... something like the deadmans switch should be implemented in a de facto utility like browser password recovery but extended with a weighted audience tied to the time elapsed. Why not have that same function that keeps up with your authentications notify X audience in Y time - 30 minutes even or 30 years with an emphasis on how perpetual. Reasonably I say we all got a good hundred years to plan for.

Sounds like an actual use case for web3. There are some blockchains like NEAR built for hosting web apps.