It is quite true that saving - alone - won't make you rich, if by "saving" one means "let the excess accumulate in a checking account."
What the article glosses over is that one needs to "save" (i.e., spend less on "stuff" than one earns) in order to buy those stocks, bonds, and real estate. That's how it worked for us: we started with negative net worth, saved, bought investments with those savings, and are now wealthy (enough).