For a single password, it is indeed illogical to distribute it to others, in order to prevent it from being stolen and misused.
That said, the concept of distributing authority amongst others is quite sound. Instead of each owner having the whole secret, they only have a portion of it, and a majority of owners need to agree in order to combine their parts and use the secret. Rather than passwords, it’s typically used for cryptographically signing off on something’s authenticity (eg software updates), where it’s known as threshold signatures:
Imagine for a moment, instead of having 1 secret key, you have 7 secret keys, of which 4 are required to cooperate in the FROST protocol to produce a signature for a given message. You can replace these numbers with some integer t (instead of 4) out of n (instead of 7).
This signature is valid for a single public key.
If fewer than t participants are dishonest, the entire protocol is secure.



Please explain what you mean by “rotate”. The thermostat is physically turning in-place, as though a wall clock?