The US dollar isn’t the largest and most liquid market on earth because the US is mean to people, if anything that would make it less useful for third parties. It’s used because it makes the most sense. Look at the USD as a gift card. If Starbucks only accepted star-bux, requiring me to buy starbux to buy Starbucks, we’d still be valuing drinks in dollars regardless of the exchange rate. Even if I could use those starbux at Target, I wouldn’t start caring about my starbux currency risk as long as I could convert to dollars. It’s the same for other countries. Japan doesn’t care about its exchange rate to the riyal because both float reasonably stably to the dollar. Saudi doesn’t want yen and japan doesn’t want to buy riyals so they split the difference and transact in dollars.
Your explanation involved a bit of mind-reading here to deduce that the Japanese don't want riyals and that people wouldn't value things in star-bux.
Going by actual evidence, if the Japanese are trading with the Saudis it is pretty clearly they don't want riyales, yen or dollars; they want goods and services that they are trading for. They would be ambivalent about what they traded for that oil in as long as they ended up with the most product out for the least yen in.
There is evidence that the Japanese want dollars (their foreign reserves are jumping), but there is no particular evidence that they want to trade in dollars. If anything it is more likely that they are trading in dollars because SWIFT is the cheapest way of moving money around internationally and it has some sort of favouritism for US dollars.
Parent is making an accounting point. Unless you have perfectly balanced trade, your balance sheet will continuously expand in either your currency, the foreign currency, or both. The Saudi government either isn’t inclined or can’t find counterparties to run enough of a trade deficit to provide the rest of the world enough riyals to satisfy their demand for Saudi oil. The USA on the other hand is willing to run such a trade deficit, it's pretty much the only game in town.
Meanwhile, the USA is able to find counterparties for its massive trade deficit because the USA itself has a huge and incredibly diversified economy and demands dollars in payment. You can buy oil from the USA in dollars for the same price you can get it from the Saudis. You can also buy all kinds of other things you can't buy from the Saudis. The dollar has massively superior optionality compared to the riyal (and most or even all other currencies) and that increases demand for it.
Another way of putting it is: dollars have a network effect going for them, and it’s stronger than Facebook’s network effect in social media. This network effect was intentionally built during the Cold War. And with the USSR’s collapse, it became that much stronger.
