You can have an income of zero and more wealth than some enormous swath of the remainder of the country. If there is one metric that does not capture wealth, it is household (or any other) income.
However, the vast majority of people earn their seed money by doing work (as opposed to inheriting it). Some but not all then amass further wealth through investment (far more these days than in bygone eras, really.) We tend to regard this, too, as a form of working for wealth, especially since so many savings plans are tied to workplaces, and since so many of us "set it and forget it," and typically we realize that wealth as an income stream.
So the kind of money most of us see makes us think our wealth results from our income. The wealthier among us understand that their accumulated and/or inherited wealth is a surplus amount above and beyond the piddling amounts that we worry about in the category of income streams.
So yes, they can have zero earned income in a year, and tremendous wealth.
Your suggested metric is better for capturing wealth at the level of us working shlubs. It's bad as a measure of wealth in general. Since the wealth pyramid is downright asymptotic, measuring this way is going to capture "what people believe is wealth" but will miss a very big chunk of wealth in the hands of comparatively few people.
The only way to look at assets is to look at assets. We use income as a proxy that's inherently flawed and that is becoming progressively more flawed with the passage of time.
The end result is reverse robin hood taxation: We only take from those for whom wealth is income; we leave alone the vast wealth that is not earned. We do not tax the super-rich on the "super" part of their wealth (i.e., most of it.)
If the median is 60k, this statement is correct. If the average (or mean) is 60K, it's incorrect. I'm not in this argument, but it looks like somebody in it introduced the oldest "whoopsie" in the actuarial book, median/mean confusion. I dunno who.