Getting started page should also state post installation steps (as well as "Hello early user," app start page)
getting started page should mention what to do after installation e.g. installing browser extensions or editor plugins. Maybe there is a way to offer installation of the browser plugin after visiting the Activity Watcher?
Since the web extension isn't really necessary, I've hidden this under the 'Activity -> Browser' tab/visualizations for now:

Perhaps it's good UX to get people to do it straight away though. There is another benefit to having more people install the extension: improve our estimates of active users (which we only get from web extension installations: https://activitywatch.net/stats/)
Could detect browser and link directly to extension store pages (most users), and also link to watcher page in the docs (for more advanced users).
yes, the more advanced/determined will find it. However, what block AW adoption is the high entry barrier.
Ideally, user after first visit of the app, should be offered a guided tour over the AW in my opinion, but that's another story.
yes, showing a google chrome store badge (https://developer.chrome.com/docs/webstore/branding/#using-the-chrome-web-store-badge) with link to google chrome store is the way to go