If you have a browser extension, you don't have to to use the search box to open another page. With an extension, it's integrated into the browser - usually as a button on the toolbar. With LastPass (and others that have an extension), I just click the button, a drop-down appears where I can select the site I want and click on it. I'm then taken to the site and immediately logged in. So yes, I can access the Google Password Manager on Firefox, but it requires the extra step of accessing that page first before I can use it.
Plus, Google already knows more about me than it has any reason to - I don't have to surrender my username and password to every site I use.
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