This is a historically ignorant argument. It's unlikely we'll actually run short of any of these metals. The more likely scenario is that if and when we do start to exhaust the most easily mined sources prices will start to rise, spurring investment in new extraction technologies and exploitation of previously uneconomical sources. Increasing production and amortization of R&D costs will then lead to lower prices in the future. This has happened innumerable times with innumerable resources throughout history.
Which is, indeed, the conclusion of the linked article as well.
Unless we stop consuming a finite resource we will eventually run out, by definition. Your argument simply demonstrates how rising prices will lead to extracting 'harder to get' resources. When something is harder to get that means it's scarce, which is the opposite of abundant. In other words, it means we are running out.
"Unless we stop consuming a finite resource we will eventually run out, by definition."
Timescales matter here. If we're going to run out of a "finite" resource in 10 billion years vs. 1,000 years or 10 years it makes all the difference in the actions you take.
Also, when something is "harder to get" it may mean merely that we haven't developed the technology to extract it yet. Aluminum used to be more precious than Gold because it was "harder to get" than Gold, though it was not scarce. And today Aluminum metal is far from scarce and far from expensive.
Which is, indeed, the conclusion of the linked article as well.