My question is, if the Treasury is shorting the dollar, shouldn’t the rest of us be shorting the dollar too?”

I'm too lazy to figure out if this is actually true, but I know I wouldn't be shorting USD right here. The US treasury market is the global safety trade (or fear trade if you want) and any crisis anywhere - China, Japan, Russia, Europe - will result in buying of US treasuries, which means demand for dollars goes up, which means the exchange rate goes up when measured in other currencies.

Remember, the treasury can always cover its position by printing more dollars. You and I can't do that.

ebear