Increasing the number of dollars floating around does not give more tax headroom. Taxes are strictly a percentage of the quantity of dollars pie. Making the pie bigger means everybody gets a bigger piece and everything increases in cost to match. Wages that do not increase result in the earner getting a smaller piece of the pie. So, the cost of labor increases as people refuse to work for an amount that doesn't pay the bills anymore. But, the percentage of the slices are exactly the same after the lag period of 12-18 months. The lag period is how long it takes an increase in money supply(inflation) to be reflected in the markets of goods and labor.
https://www.johnlocke.org/the-myth-of-wage-push-inflation/
https://www.lancaster.ac.uk/staff/ecajt/inflation%20lags%20m...
Even the small capitalist class benefits from the heated-up economy, because the increase in wages results in money being siphoned up from the pre-existing stockpiles of the wealthy, which is where money tends to fall when left to its own devices. When money gets siphoned out of stockpiles and up to people who have to spend it to live, the rise in effective demand makes it valuable to invest in providing more supply, instead of just sitting on the money and collecting interest.
The one major downside to wage-driven inflation is that it hits people living on fixed incomes in the shorts, since it makes pensions and government-provided supports less valuable. This is not a reason to support a heated-up, wage-driven-inflation-powered economy -- it's a reason to support government measures to increase funding for the elderly and disabled in order to match the new value of money.