The Apple / Google contact tracing app doesn't store or use any location information, and very cleverly protects privacy. That's the system mostly used in the US.
If a government decides not to use that method and to understand do location tracking, yeah they just might have a motive for that decision. :)
this is how the Singapore app works too. When you register the app (or the standalone bluetooth token), you give the Ministry of Health your contact details, but the phone apps and tokens only swap your random ID with each other, and those exchanges stay on the device.
If you are diagnosed with Covid-19, the MOH gives you a code you put into the phone app that will let it upload all the recent exchange data to their website, where it can be associated with your contact details. (Or you give them the bluetooth token, if you are using that instead of the phone app.)
If the police have access to the MOH registry, then they can only access people's contact details (which the Singapore government already knows for everyone in the country), and presumbly the uploaded random identifiers for anyone unlikely enough to get a Covid-19 diagnosis.
There is currently a separate system called SafeEntry which is based on scanning a QR code when you enter any shop or shopping mall, and presumably that uploads identifiable location data to a government website. However if the police are really interested then they can already access cell phone tower data to know where your phone has been. (even pre-pay SIM cards in Singapore require ID to purchase, so the government knows who has every number.)