Check out Mull or Mulch. Mull is a more private Firefox and Mulch is a more private Chrome. They are made by the DivestOS team, they also make DivestOS which is a lot more private Android version that supports a huge number of phones. That’s how I’m also runing the latest version of Android 13 with the latest security updates on a phone from 2019. And there are much older devices supported.
Yes and they apply while the device is off too.