The infringement is not running the game in the emulator, it's obtaining an illegal copy. The Betamax case you cite is about devices which have this capability: with a recorder you can obtain an illegal copy of a copyrighted work. That's not the case with an emulator: an emulator merely emulates a given piece of hardware and runs whatever software you load on it like the original hardware. It's not capable of providing illegal copies of copyrighted works by itself, the illegal copy has to be obtained in some other mean.It's a straightforward example of contributory infringement. The only real defense would be if they could establish that the emulator has "substantial non-infringing uses" as established in the Betamax case. I don't think there is a case for that here.
Basically, I don't see any particular feature of this emulator which contributes or facilitates getting illegal games, it merely allows you to run them if you already have them (which doesn't make it illegal). The only thing remotely comparable is providing the in-app web browser, but it's still the end user which has to decide whether it uses the browser to download ROMs illegally or read game guides on gamefaqs... let's say that I doubt it would be considered "without substantial non-infringing uses". A different thing would be the emulator providing some sort of direct "in-app download" of illegal ROMs. As far as I understand that's not the case.
The wikipedia page about console emulators says this:
Although I have no idea what these legal precedents say exactly.According to all legal precedents, emulation is legal within the United States.