[RedBeetDeadpool]

Because there are no dollars involved. If they do as you suggest, at the start:

A holds gold.

B holds dollars.

C holds oil.

A trades B, gold for dollars. A holds dollars, B holds gold.

A trades C, dollars for oil. A holds oil, C holds dollars.

After your suggested trade result:

A holds oil.

B holds gold.

C holds dollars.

If there is no "middle man", party A gets oil, party C gets gold. Party B keeps their dollars. End result:

A holds oil.

B holds dollars.

C holds gold.

Assuming "B" is USA, USA doesn't get to export its inflation/funding for stimulus checks/student debt/pension crisis/(or in trump era - a wall that does nothing) away to "C", whoever that ends up being, meanwhile, Ghana gets the oil it wants, and "C" gets currency without having to pay for the choices of politicians they have no control over.

[nickdothutton]

The US export of inflation via engineered demand for petrodollars (and one could say enforcement via aircraft carrier groups) is an under appreciated effect.

[luciusdomitius]

It is literally the most controversial concept in today's geopolitics. Study politics science in Dover - it is a conspiracy theory. Study PS in Calais - it is the exorbitant privilege and a cornerstone of global injustice. And it is not like France and the empire are rivals.

https://en.m.wikipedia.org/wiki/Exorbitant_privilege

[happyjack]

This! This is the answer! Trading in dollars creates micro transactions where the USA can export inflation and much of its problems.