Not necessarily!
If you implement a tiered tax credit on gross regenue earned for wages paid, that decreases exponentially from the lowest-paid worker percentile to the highest, then producers could offset the tax on imported materials by paying an increased share of the revenue to their lowest-paid workers rather than raising prices. The government wins because household buying power and taxes paid increase, the household wins because buying power increases are decoupled from price increases, and shareholders win because wage increases shift the domestic demand outward (rightward), compensating for downward shift along the demand curve from price increases.
Setting the refund rate to (the target inflation rate minus the current inflation rate) times a constant defined by the Fed alongside the reserve rate also provides the Fed a long-term lever of financial incentive: for firms to increase wages paid rather than profits paid, when the Fed raises the constant to combat price inflation, and to increase profits paid rather than wages paid when the Fed lowers the constant to combat price stagnation. It also offers a control against layoffs for profit by specifying such that reduction in workforce in each percentile reduces the target inflation rate in the above equation; a layoff of the entire lowest-paid percentile to avoid lowering prices would result in a significant tax penalty charged at the next higher percentile’s rates, unless prices were so stagnant (or decreasing!) that the Fed’s target had been adjusted to allow it.
That this isn’t coded into today’s U.S. monetary policy is certainly true; but it merely requires an act of Congress to resolve. It’s important not to take for granted that what we’re used to is therefore what must be.