Yeah, it’s rare that 1P fails to recognise the website, I always get suspicious. Most of the time when it fails it’s on random crappily made niche websites anyway, for which I wouldn’t care if my credentials were stolen!
Actually, when I do get worried is when I have to login in an app with a credential I saved for the website. 1P doesn’t match the app automatically (understandably) and I’m always a little uncomfortable