Their Google dependency is their existential problem. They're limited by what they can do with "making Firefox better" while effectively being a client state. An off the books Google department. Doomed to forever being a worse funded Chrome because they can't do too much to anger their patron.
By selling browser UI real estate to AI companies[0] they reduce the power Google has over them. If they get to the point where no individual company makes up a majority of their revenue, it allows them to focus on their mission in a much broader way.
[0]These will be very expensive listings should this feature become popular: https://assets-prod.sumo.prod.webservices.mozgcp.net/media/u...
Is there any prove for Googles influence on their development you outline here?
Yeah but is this entirely true though? It seems Google pays FF just for existing, to protect them from antitrust litigation (or what's left of it); so Google can't really stop paying FF and can't try to kill it, as its death would be extremely counter productive. FF may be freer than it thinks.