I must be missing something important, hopefully someone can explain this to me.
If the Saudis agree to price their oil in dollars, that essentially places their oil in the same situation as goods produced in America. If the money supply doubles, the price of American goods double, including Saudi oil. This is no different than if the Saudis didn’t price their goods in dollars but in some other currency, since a doubling of the money supply would cause the exchange rate of the dollar to fall by half. Either way the price of oil would double in dollar terms. So there is definitely no sense in which the petrodollar allows inflation to be exported.
The only way inflation could be exported is if the price of oil in dollars was fixed. In that case, if the US money supply is expanded, oil would fail to rise proportionately with other goods priced in dollars, causing the relative price of oil to fall. More oil would be imported into the US, dollars would flow outwards (into foreign reserves), and American prices would rise less than otherwise would have. (This is why fixed exchange rate systems cause inflation or deflation to be transferred between countries.)
As to the question of whether the petrodollar raises global demand for US dollars, why this is important baffles me. After all, dollars are merely paper, the real value of American products hasn’t changed. If the Canadians decided they would adopt the US dollar as their currency, the result would be deflation as there would be the same quantity of US dollars spread across a greater amount of production. Sure, the dollar would be more valuable, but that rise in the value of the dollar would be exactly offset by a fall in the nominal incomes of Americans.
"The Petrodollar" was a creation of the diplomats, not economists. It was a project of Kissinger and Nixon as a weapon against the Soviets and the "3rd-world", aka non-aligned countries.
"Raising the global demand for dollars" has the effect of forcing increased reliance upon US (dollar)-based financial institutions to settle transactions, and that means that the US Government (through its regulatory system) can exert control over the transactions of 3rd parties, otherwise known as "sanctions".
If you need to use dollars to purchase some good on the global markets, and the US Government says “nobody using dollars can do business with you”, you’re suddenly going to have a hard time finding goods you are able to purchase. Even worse, your foreign reserves, deposited in a dollar-denominated account, would also be frozen and suddenly unavailable to make the purchase, even if you could find a seller.
Macro-economic analysis is not going to get you anywhere, because it’s not an economic system; it’s a system of power and control.