If they get you using their browser they have complete visibility and control over what you do, whereas Google and Apple control the dominant browsers and subsequently what anyone else can build upon them.

Google has also locked-down default search engine on every major browser, including in at least Apple's case contractual obligation not to provide a search engine selection screen so they're fighting an up-hill battle for adoption there.

This doesn't allow for much and I think especially in terms of agent-control, for example IIUC Playwright and Puppeteer have always had to use modified derivatives of the main browsers for their automation.

it was chat-ui last year, vscode forks this year and maybe browsers are next.

I think they are also building an IDE too. It's just about control.