Again, that's a very intriguing question.
As I see it, the current problem is that making the app mandatory would contravene a number of laws, specifically in relation to consent. Use of an application usually requires people's consent. What you're proposing, however, would require people to share their personal information. That strikes me as very difficult to do in a country like Canada.
In terms of how Quebec law pertains to public organizations, consent poses a problem there as well. As someone who's not a public health expert, I have a much harder time understanding how something like that could work.