Is there anything stopping the NYSE, Nasdaq or CME/CBOT from handling trades in another currency?
It would reduce liquidity. Equity prices would fluctuate not only on buy/sell basis but exchange rates. Sure, computers could figure all that out these days but what's the advantage? Overwhelmingly, equity buyers and sellers (not "traders") buy in their local currency because they use the money to live in a local economy.
Companies list in other countries for access to those countries' buyers. What would be the point of Shell listing in Euro on the NYSE? They want to list in dollars. Nobody outside Nigeria lists on its exchange but local companies do because local people understand the companies and everything (both their operations and their stock) is in naira.
So if you want to be an exchange in a different currency, just buy a local exchange. NASDAQ did try to buy the London Stock Exchange, though I think it fell through.
If countries then acquire dollar surpluses by running trade surpluses with the US, the US by contrast has a trade deficit. This is equivalent to having a capital surplus for the US. It means excess capital is funneled back into the US into the capital markets buying stocks and bonds.
This is maybe a chicken and egg phenomenon..is it the demand to invest in the US creating a capital surplus that creates the dynamic whereby the $ becomes reserve currency and the US runs increasingly large trade deficits? Is it the military/political power that creates all of the rest? Probably all of above. But in any case the reserve currency system has at its core the financial markets of the US that the rest of world invests their surplus into, incentivizing them to produce in excess and trade real goods and work with US in exchange for paper IOU's that they can invest into the US markets.
A big aspect of this $ financial/trade system isn't just the $ as currency itself but the unique and important position of US treasury debt as the premier reserve asset that countries store their surplus and forex reserve in, and which is the center piece of the eurodollar[0] lending markets.
The policies you describe as "idiotic" are rationally imposed. The assumption is that they will weaken Russia's war effort, and that the cost, while high, is much lower than fighting a hot war down the road if Russia is allowed to continue to invade its neighbours.
You can argue that the policies aren't working, but while looks like that in the headlines, if you look at what's going on inside Russia all the lines are pointing down, even if the government's own figures claim otherwise. You could argue that a slightly different class of restrictions could be more effective.
But the only basis for "idiotic" is if you think it's none of the EU's business if Russia chooses to invade and try to conquer one of their neighbors.
Unlike the chicken/egg situation we know precisely when this system was born: the Bretton Woods Conference, 1944.
To some degree of course it recognized what was happening anyway (uh oh, chicken/egg is back) but rather than letting things evolve it built upon emerging practice to build the modern global financial system (basically still in place despite further evolution, like floating currencies).
Also, the US stock market, US Dollar and US economy/gdp aren't hard linked to one another these days. The companies listed can be selling to non US markets, employing internationally, founded internationally. They're just listing on the US stock market because well.. that's where the stock market is. The US could, in theory, become more or less popular a stock market regardless of its currency's popularity.
Meanwhile, both the Euro and RMB have similar size markets backing their currency. Neither one is currently trying to displace the USD. I think the importance of owning the international currency is somewhat speculative.