Are you sure this is what your local app does? Many COVID-19 government apps were built reflecting this desire for privacy, I've written about the New Zealand one previously but lots are like this.

When you scan a QR code with that Kiwi app your phone learns you went somewhere and when ("This code is for the Auckland central library, and it's 1430 on Tuesday 16 March") but it doesn't tell the government, they don't care and could only make things worse by losing the information. It just remembers where you were.

Then when the government finds out that an infected person was careful to stay home except, oh yeah, they did pop to that library to get a book to read while they stayed home, for about 15 minutes, around 2-3pm on Tuesday, they send all those apps a message (it also goes in a press release but who seriously reads those?) and the app goes "Auckland central library? 1400 to 1500 on 16 March? That's a bingo" - and you get a message telling you that you should get tested, or to watch out for symptoms or whatever the government advice is in that particular case.

So effectively your phone is just simplifying work you'd otherwise have to do, instead of you laboriously checking the list of locations in your local paper or on a web page any time there's a breach, the phone matches it correctly for you.

If you're infected, you do have the option to have your phone tell the tracing people everywhere it remembers you going recently, but that's up to you whether you feel morally obligated to help them. Contact tracers in countries with low incidence are mostly from STI clinic backgrounds (which of course also need tracing), so "I went to the restaurant even though I had virus symptoms" is at least easier to confess than "I fucked some random stranger I met in a bar last Tuesday even though I'm married"