You are correct in thinking that greater dollar printing does not necessarily result in greater adoption of the dollar. However, that is not the point in the explanation you have read.
The point there is that compared to another currency the dollar is more resistant to inflation from printing than others are, because there is more foreign demand for dollars than there is foreign demand for other currencies.
More dollars only lead to inflation in the U.S. if the dollars are actually there, otherwise they may just as well not have been printed as far as the domestic market is concerned. Since the dollar is a famous reserve currency and is held in significant amounts in many other contries, the relationship between dollar printing and inflation is different compared to other currencies.
A numerical example might illustrate the point:
Assume 1000 units more of a domestic currency lead to a 10% increase in inflation in that country. This is true for all countries in our example.
If e.g. Mexico prints 1000 pesos then all of those pesos will stay in Mexico (no foreing demand). So inflation in Mexico will be 10%.
If the U.S. prints 1000 dollars, 500 of those go abroad. So the U.S. has 500 more dollars domestically. So inflation will be 5%.
So if the U.S. prints 1000 units of currency inflation is 5%, but if Mexico prints 1000 units of currency, inflation is 10%. In conclusion the U.S. has a different, relationship of printing to inflation. Specifically it has less inflation, i.e. is more resistant to inflation compared to other countries.