Just don't try and use it with Edge - on my work Ubuntu laptop, the PWA is a steaming pile - locks up, crashes, eats all the CPU - I gave up and just have it as a pinned tab now.
I'm not sure the PWA would bring any privacy advantage?
Potentially limiting what the app has access to. A "real" app can see your clipboard, files on your disk, and the content of your browser profile (technically a subset of "files" but worth calling out on its own). A web app can "only" see your interactions with the web app and data in it (which in this case includes all your email, so...).
I see the point concerning the limitations of what a PWA can do, but as here it's a question on snooping on emails... it probably wouldn't make any difference on that matter.