You cannot run Signal without "Signal - the company" existing. All of their systems are designed to be attached to one specific backend, namely the signal-run backend, meaning without re-engineering the existing infrastructure you cannot simply swap over.

As @kpw already mentioned, "Signal - the company" dying would involve a functional reset of everything: No contacts, no servers, no infrastructure. COULD you fork the thing and build you own system? Sure, but it would be functionally unusable since no one else would be using it, since everything relies on specifically the signal servers to function. A post-signal system could re-use some of their code (if it runs outside signal corp - "works on my machine" could be present in this project as well), but would need to rebuild the actual network.

This is in contrast to something like the matrix protocol: If a specific matrix instance goes kaput, you still have the overall network working. This means that even if an instance implodes, you would have an easy migration path since the matrix network itself persists.