I think you've done a pretty good job of separating out the strands. "Money can't buy happiness" is just another platitude, IMO.

What I believe is that the people who say it are trying to home in on the fact that the desire for money is a fairly miserable desire. If it's frustrated, you're unhappy, and even if it's satisfied, you're not necessarily a lot happier because you may have sacrificed too much to get it.

But the lack of money, as you have pointed out, can cause a wide variety of miseries.