How? It doesn't change anything with regard to this specific concern about family. They already have an idea about how you live and how much you have from knowing you. They might not have a specific dollar amount, but they'll have a good idea.
That said, I don't think it should be necessary to defend privacy. The idea proposed up-thread is to publish all income data on everybody. Not merely to encourage people to share their income data so they can reduce the information differential with employers, or some central opt-in system to make it easier to compare the numbers. It's about fully _enforced_ transparency.
A proposal for such forced loss of agency needs explanation and must be defensible against scrutiny, not the request to leave people alone just like they're used to.
> But have you thought about why you feel this way?
Not every family is healthy. Some become _decidedly_ unhealthy as soon as money comes into play, and having precise figures available can be really destructive.
It might work better in more egalitarian societies, where the difference in pay is less to begin with (eg Scandinavia). But that's not a road the US (and many other places) is willing to take.