'Aarogya Setu' app launched by government works exactly as you are suggesting. Data of citizens are filled on the app and it will alert you if someone with Coronavirus symptoms come close to you. However, an ethical hacker questioned the security of app last week to which Aarogya Setu team replied that the app is completely safe.
The security and confidence can be there as long as the details are not leaked, we experienced the huge leakage of aadhar card in the past and the government was ensuring everything is safe but millions of cards were available online.
Although the questions raised by hacker aren't related to personal data but still I think introduction of blockchain based database will help in complete security of personal information.
Introducing a blockchain can help in the safety of personal information, like tampering of our data or someone deleting our data but it does not give you the guarantee that the data will not be leaked.
Anyway, it's a need for now to have the app for your own security i guess
If you are not roaming outside you really do not need a surveillance system attached to you. Since it tracks the location and if you are in a major city and living in a big society it is possible that it will give false positives to the government.
The thing that made me think about the app is
" Aarogya Setu tries to determine the risk if one has been near (within six feet of) a COVID-19 - infected person, by scanning through a database of known cases across India."If someone is infected and later identified to be a positive case then it will give the government the information about anyone who had access or crossed path with the infected person. I hope this is what they meant by this statement.