I have it installed as a home screen app. There's no white flash in Safari, but a flash does occur when you launch the installed app.
iPhone 13, iOS 16.5.1 here, and I get the white flash too when installed as a PWA.
Maybe it's an iPhone thing? On my s20 it doesn't do that.
Very likely is an iPhone thing, but I'm hoping this can be fixed since this is definitely going to be popular with ex-Apollo users.
It happens with my other PWA too.