Most rich folks aren't rich from holding dollars, they're exposed (to your point) to equity. Higher expected inflation usually means higher future nominal interest rates, and a higher future cost of capital. The DCF model used to value that equity means they get put through the ringer in exactly the way you're seeing paying out in the stock market right now.

If a company gets a loan while interest rates are low, then does a stock buyback with cash on hand, isn't it the same as using the loan money?