I suspect Google would prefer a zero-click solution for their browser.
You have to put in your email address and click a button still. There's a video demo of this in TFA. There's no way they've been dumb enough to let websites figure out your email address automatically, come on.
your browser accessing your inbox
Accessing your email via your web browser is literally how 99.99% of people use email. There's nothing dubious about it.
I get that you have a gut fear about it though.