Noob question: passwords are exposed to agents, right?

Forgive me if this is a beaten-to-death horse, I'm new to all this: If you use e.g. Claude Code to automate some business tasks, and you manually enter all the passwords on the websites that Claude needs access to every time those tasks are working, and those passwords are obscured and not visually revealed, the connector module in Chrome can nonetheless still see the passwords via the DOM, right? I understand that the agent will be prompted not to ever do that (at least, we hope so and trust a gigantic AI corporation to do so), but this seems like a pretty significant security lapse that something like 99% of people who are using these agents are unaware of. If Claude is itself hacked, or a prompt injection somehow happens, or the LLM just gets a little confused, it could be sending the password back to the cloud, yeah? I understand that people trust big corporations with their passwords (e.g. microsoft), but a hyperscaling AI startup seems like a sketchy org to be trusting with the keys to your kingdoms... especially since every state and criminal actor in the world would love to use their own AI to hack in and see what they're capturing... am I off base here?