Nintendo can't prove the leak wouldn't have happened without Yuzu. But they don't have to. They can simply show that Yuzu made it infinitely worse. Also, suing Yuzu does not preclude suing emulator makers or flashcart makers - they can all be sued as they all have culpability.
One thing to take into account is that civil trials don't require the absolute highest standards of proof, such as proof beyond a reasonable doubt. Usually much lower evidentiary standards such as "preponderance of the evidence" and other malarky.
The law doesn't work on strict abstract principles. Windows has a million uses of which piracy is just one. Yuzu has two uses (playing legitimate backups and pirated backups), of which Nintendo may successfully argue, both are piracy (due to the circumvention of encryption in both cases). In which case, the only possible use in 99.9%+ of cases... is for illegal activity.
I don't think it's likely that software developers can be sued on the grounds of "some people use it for illegal activity". Can IP rights holders sue any torrent tracker on that basis? Not to mention that playing backups of your own games is explicitly legal in many places, and I'd be shocked if it isn't in the US. Lastly, one can make an argument that the purpose of Yuzu isn't playing "Switch games", but emulating the Switch hardware stack, which is legal. For example, writing your own Switch homebrew and running it on Yuzu is permitted.
