The website is missing the password, and the recipients don't have the file until the website gives it to them.
(Clearly this still isn't (and can't ever be) sufficient though: bribery/threats of violence/blackmail by either party towards the other.)
Alternatively, let the dead man's switch alert heirs to swap their half of the password, so that all heirs now have the full password. Then you don't overly depend on the dead man's switch.
Once it earns some trust, it could offer to sell your startup/website/app to someone that would do a good job of maintaining it (or release it as open source).
As it even notes on the page its 'running on free quota' - hardly something to actully be trusted with life or death information.
A cool tech demo, not a 'premium' service.
Like you note with your current setup, the passphrase is useless on its own.
Or store encypted file in one deadman.io switch, and the passphrase in another switch (in another account). The evil owner of said website, shouldnt be able to connect the two. (whats the chances they going to arbitarly try decoding every file they have with every passphrase they have)
I think the ultimate solution is a dead man's switch that is tied to something physical, aka a password in a bank vault that unencrypts a file somewhere. Yes, you have to trust the bank, but it's unlikely the bank knows what to do with this random password.
This is what I do - all my passwords and super-secret stuff is in an encrypted file with the passphrase something I have memorized. It's also written down and stored in a secure location that my wife has access to, should something bad happen to me. I don't care if she accesses this password when I'm not dead - it's only in a secure location to prevent accidental disclosure (theft) of it.