As we have observed, there are no natural units for the measurement of purchasing power, and there is no natural landmark to tell us that one purchasing power is better than another, either. It probably doesn't, but most economists would say that even though the price level doesn't matter, changes in the price level do matter. It creates uncertainty, in that people do not know what the money they earn today will buy tomorrow. Uncertainty, in turn, discourages productive activity, saving and investing.
Inflation reduces the competitiveness of the country in international trade.
If this is not offset by a devaluation of the national currency against other currencies, it makes the country's exports less attractive, and makes imports into the country more attractive, which in turn tends to create unbalance in trade. That is, people who hold bonds and bank accounts in dollars lose the value of those accounts when the price level rises, just as if their money had been taxed away. As the purchasing power of the monetary unit becomes less predictable, people resort to other means to carry out their business, means which use up resources and are inefficient. These inconveniences might be reason enough to call for a stable price level.