What advantage does having reserve currency status bring for US dollars in comparison to other foreign currencies?
There are several advantages of being country that issues reserve currency:
- Increase in seignorage revenue for issuer. For example, there are some estimates that show that $10\%$ increase in Euro reserves (Euro is one of the main reserve currencies beside dollar) can lead to extra seignorage revenue of about $0.5\%$ GDP (see Blanchard et al Macroeconomics pp 541).
- Country benefits from less exchange rate risk, since trade is sometimes settled in terms of reserve currencies circumventing forex market. (see Obstfeld 2019).
- It lowers country's borrowing costs (Warnock, 2010).
- The extra demand on forex markets rises dollars purchasing power
There might be some other benefits but I think the above summarises the major ones.
Is there a more formal economic theory(or school of thought) that has a similar narrative and what are some common objections?
I am not aware of any serious economists having such narrative. There is no evidence that having reserve currency would make 'printing your way out of recession' more viable. That does not even make sense.
Monetary policy stimulates (open) economy in two ways. First, by rising prices since typically high unemployment is and low output is a result of nominal rigidities such as sticky wages (see discussion in Romer Advanced Macroeconomics ch 6). Inflation forces people to take real cut from their wages even if their nominal wages remain same. Next in open economy monetary expansion makes currency depreciate which in stimulates exports because it makes country's products relatively more cheaper to that of their trading partners (ceteris paribus).
This is in no way impeded by country not being an issuer of world's reserve currency. There are some benefits of issuing reserve currency when it comes to monetary policy, but as explained in Obstfeld (2019), the main one is that it allows the central bank to affect and help to stabilize global asset prices. So it means that such country can engage in macroprudential policy on more global scale. In addition, as discussed in the same source having reserve currency puts downward pressure on import prices. However, none of these make expansionary monetary policy more viable per se. Being able to do macroprudential policy on international scale is helpful but mainly to the extent that the crisis country faces is global. Downward pressure on import prices might actually even lead to less stimulation of an economy, as that means imports won't fall as much so some of the extra aggregate demand will be directed toward foreign products and stimulate foreign economy (although it is not end of the world either since increase in output in foreign economy will also increase their respective demand for both domestic and foreign goods).