The interesting thing about ideas like this is that they reveal how authoritarianism during times of stability does a terrible amount of damage during times of crisis. This is the sort of problem that could be solved in a better world; collecting location data to find interactions between people would allow you to get them tested faster and prevent a disease from spreading. But it simply can't be done in this country…
I’ve regressed to my cynical default in the past few days, but “a large institution that I would trust with this authority” sounds like pure fantasy.
You can mitigate this to some extent by having one or a couple people who are personally responsible for the whole organization (royal families in monarchies, the owners of privately owned companies, etc) but then your organixation is not resilient to bad leadership.