From a technical perspective, you don't need to trust the government in these cases - provided that they implement the solutions with built-in privacy.
Singapore (and others, for that matter) has allegedly solved some of these issues in their soon-to-be open source contact tracing app [0].
They basically let every device keep track on itself and it's encounters - until a diagnosis is made.
Can't remember the details from there off the top of my head, but you'd either do a lookup via a central authority, or notify peer-to-peer, depending on what other mechanisms are in play (ephemeral/co-signed IDs, etc.)