Other countries who are not reserve currencies have higher purchasing power than USD. So if the US wasn't the reserve currency, there is no reason it couldn't do the same.
You're making a claim that you haven't backed up.
Fact: The USD has large foreign demand which does not exist for other currencies.
Fact: If more currency is demanded, it strengthens the currency in a relative sense.
Fact: Therefore the USD is stronger than it would be otherwise, considering only local demand.
Your whole counterargument seems to be based on a misunderstanding of real vs nominal. To say 1 Euro > 1 USD means people in Euro earning countries have higher purchasing power is simply wrong. All currency values are nominal.
Fact: If USD money supply doubled overnight and all holders got 2x the amount and wages/prices adjusted automatically, the Euro would appreciate 2x against the dollar.
Fact: That does not mean Euro earning citizens have higher purchasing power. It's an entirely nominal change. 1 hour of labor in USD terms earns the same in purchasing power, regardless of the face value of currency. Same concept as a stock split
Fact: If Euro became the reserve currency tomorrow, Euro-holders would have more purchasing power due to foreign demand making each unit of currency more valuable in a relative sense than it was previously.
Fact: If USD lost reserve currency status, it would weaken relative to other currencies.
None of these are disputed, but you can believe whatever you want.
The actual nominal conversion rate of the currency is pretty meaningless. The fact that Yen is 100 to 1ish to the dollar is also not very meaningful. They're all just nominal values. Only the exchange rate relative to fundamentals matters. USD gets stronger exchange rates relative to the fundamentals of the currency (because many foreign countries and citizens also hold USD, not just locals). Which is mostly untrue for other currencies.
PPP adjusted salaries are lower everywhere else in the world, partially, but not wholly for this reason.