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Using an E2E chat app in your browser necessarily makes the keys and decrypted messages available to your browser. They would have the ability to read messages, impersonate users, alter messages, etc. It would defeat the purpose of a secure messaging platform.
“They” is the browser/browser maker. The browser, acting as the client, would have access to the keys and data. The browser maker could do whatever they want with it.
To be clear, I’m not saying they would, only that it defeats the purpose of an E2E chat, where your goal is to minimize/eliminate the possibility of snooping.