Firefox literally sends your keystrokes to Google, right out of the box. Brave, however, was found to be the most private popular browser by reputable researchers: https://www.scss.tcd.ie/Doug.Leith/pubs/browser_privacy.pdf.
Brave [never] hijacked links either. Affiliate Links were offered among suggested sites for relevant search input. So if you searched "Binance," the browser would offer (among other suggestions), an affiliate link for the site. Users could then choose to browse to the property with the affiliate link, and in so doing support the development of Brave. No impact to privacy or security at all.
The mistake here was with input handling. Built to handle search input, this feature also mistakenly handled fully-qualified domains. While we intended the app to offer affiliate links (when relevant) to something like "what is binance?", it was also offering them for "binance.us". The latter case was corrected quickly (and the feature itself was disabled out of the box).
More about that on our blog: https://brave.com/referral-codes-in-suggested-sites/.
To your second point, about setting up crypto wallets and soliciting donations on behalf of non-participating publishers, you're mistaken there as well.
To prime the support-system in Brave (called Brave Payments at the time), we staked Brave users with tokens, inviting them to direct those tokens to creators they would like to support. More clearly, Brave gave Brave users say over where Brave ought to direct its own tokens.
Unfortunately, our UI/UX wasn't very clear about which creators were verified, and which were not (we followed the Twitter approach, marking verified creators with a checkmark, but doing nothing for others). This resulted in some confusion at the end of 2018, where users were directing Brave's tokens to non-participating creators (most notably Tom Scott).
We received considerably helpful feedback about how the system could be improved (both from a UI/UX side, and operationally). Frankly, I don't think I've ever seen our team work so hard, and churn out such a monumental update in so little time. We had made massive changes within 48 hours IIRC. Creators were explicitly marked as verified or unverified in all cases, the BAT that Brave stakes with users would remain in the local wallet until it could be received by a verified creator. And BAT that sat pending for 90 days would be unlocked again for the user to direct elsewhere.
Tom Scott was kind enough to review our changes, and explicitly gave us his approval soon-thereafter. What is now 'Brave Rewards' wouldn't be doing so well today were it not for Tom and so many other incredible users helping us find the best path forward.
More about that on our blog: https://brave.com/rewards-update/