US printing is not the same as everyone else printing. Almost all trade deals worldwide are done in USD. Everyone except the US must export goods and services (i.e. actually work) to acquire the dollars which will finance their imports; US can just print.
Goods and services that are traded internationally are priced in USD. Other currencies do not end up directly representing goods and services; they are relative to the USD. Therefore, the demand for dollars is always very high, and it will be as long as the dollar keeps the reserve currency status.
Goods and services traded internationally are priced in whatever the parties want.
Ultimately they want to exchange for the currencies they need to pay bills, wages, and taxes in.
"Reserve" currency has little to do with it - if anything it's the high volume of trade with USA that makes it worthwhile to keep reserves of USD, because the high volume of US trade means you get many chances to exchange USD for whatever currency you actually need.
Goods and services that are traded internationally are priced in USD. Other currencies do not end up directly representing goods and services; they are relative to the USD. Therefore, the demand for dollars is always very high, and it will be as long as the dollar keeps the reserve currency status.