This is one of those rare cases where you have to directly balance lives against privacy. My feeling is that while privacy is important, it's being taken a bit too seriously given the severity of the crisis. Google and Facebook et al. carry out far more involved and intimate surveillance of people's lives than would be required for an app as described in the article.
This is not an all-or-nothing scenario. The choice is not between an app that saves lives and compromises privacy, or no app and lives lost. Privacy preserving technologies exist, and now is the time to use them. There is an additional technological cost, but that's what we should weigh the privacy costs against. The choice is between an app that doesn't care about privacy versus one that does.
Time is of the essence here. I agree that, all else being equal, privacy should be respected. However, if it takes multiple weeks to iron out all the potential privacy issues, this approach becomes much less effective.