I was referring to the Discourse hamburger menu (not the chrome menu). Actually, now I see the android screen shots were also from the chrome menu, not the discourse hamburger menu.
It would be nice to have the button in the discourse hamburger menu–the user would see it more often, and it would be a little clearer in terms of how to instruct users to install as a PWA. Probably not worth a huge developer effort, but would be a helpful touch.