Except that you've got a Federal Reserve actively ensuring that there aren't too many (or too few) dollars in circulation for the level of interest rate they want to maintain. They do it by buying and selling bonds rather than stocks, but the principle is the same. So we don't have to worry about shifts in portfolio preferences leading to an unexpected oversupply or undersupply of dollars

But more people wanting to hold AAPL instead of hold cash absolutely will reduce the amount of AAPL stock your $1k is able to buy, of course

Maintaining an interest rate isn't the same as maintaining a fixed rate of CPI growth, especially not in the short run.

Sure, but we're not talking about fixing the rate of CPI growth (stock prices aren't a constituent part of that and don't have a fixed relationship with it).

We're talking about the effect of wealth holders wanting more of their investments in stocks or more of them in cash on the USD. An increase in demand for the liquidity of currency holdings, or an increase in demand for stocks due to perceived improved returns could affect the dollar through knockon effects on availability of credit, but the interest rate for that credit is the price of the bit the Fed fixes.

Change in preferring the Yen or the Euro, maybe, but I've never read/seen anything that would reinforce the idea that the dollar would be weakened by preference for holding other assets whose values are ultimately figured by their worth _in dollars_.