Earlier quoted context omitted.
Brave Search [1] is 100% independent. There's also Yandex [2] which also works excellently, but is biased towards more Russian language results. The image search is second to none though. [1] - https://search.brave.com/ [2] - https://yandex.com/
Brave search is only very recently independent, and not, as far as I can tell, 100% of the way there. While they are building out their own index, they only recently stopped augmenting their own results with Bing, and are still augmenting them with Google.
The “augmentation with Google” that you mention is an optional feature that only works in Brave browser and doesn’t mean Brave Search is less independent than in other browsers (but it might mean that the quality is increased for some queries; or at least closer to Google results in these cases).
I think when talking about independence it is important to make a distinction between “dependent” which means “cannot work at all if the provider goes down” (e.g. yesterday’s outage) and “dependent” meaning that the service would continue to operate but with some amount of degradation of service. It’s a spectrum. Yesterday we saw that some services could not operate at all when Bing was down.
Disclaimer: I work at Brave.