That's incorrect. You need your phone to log in, but you only have to sign in once per browser. (I assume sessions time out at some point. You can also invalidate existing sessions in the app.)

It's correct that the phone does have to be online while you're using it. But at least you don't lose encryption. Somehow, Signal's web-based client manages to do e2e without that requirement; maybe there's a non-technical reason why WhatsApp does it the way they do.