Absolutely. However, it is easier to start with just Chromium flavor of browsers.

The two important concepts from Puppeteer/Playwright are managing the lifecycle of pages (tabs) and frames and the other is using handles / locators.

There are a lot of limitations using the extension API in any browser instead of CDP / Webdriver BiDi. I'm curious, how would you use this idea?

Could be cool for automated website UI testing/verification (check if certain text appears), perhaps also with network API events as well.