Earlier quoted context omitted.
The mere act of browsing the web is "write permissions". If I visit example.com/ , I've now written my password into the web server logs of that site. So the only remaining question is whether I can be tricked/coerced into doing so. I do tend to think this risk is somewhat mitigated if you have a whitelist of allowed domains that the claw can make HTTP requests to. But I haven't seen many people doing this.
The thought that occurs to me is, the action here that actually needs gating is maybe not the web browsing: it's accessing credentials. That should be relatively easy to gate off behind human approval! I'd also point out this a place where 2FA/MFA might be super helpful. Your phone or whatever is already going to alert you. There's a little bit of a challenge in being confident your bot isn't being tricked, in ascert…
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.