You do need to gate the web browsing. 2FA and/or credential storage helps with passwords, but it doesn't help with other private information. If the claw is currently, or was recently, working with any files on your computer or any of your personal online accounts, then the contents of those files/webpages are in the model context. So a simple HTTP request to example.com/<base64(personal info)> presents the exact same risk.

You can take whatever risks you feel are acceptable for your personal usage - probably nobody cares enough to target an effective prompt-injection attack against you. But corporations? I would bet a large sum of money that within the next few years we will be hearing multiple stories about data breaches caused by this exact vulnerability, due to employees being lazy about limiting the claw's ability to browse the web.