No, you shouldn't use Brave.
The Brave browser is a fork of Chromium with a built-in adblocker and some other neat privacy-friendly features. From a quick glance, it looks amazing, right? Chrome but pro-privacy! Sadly, you'd be wrong. Brave's history is full of controversies, many of which are deal-breakers for people.
1/11/2021 edit: Many people seem to think that this article is just "Brandon Eich bad" or that I don't like Brave because I don't like Eich. That's simply not true; Eich's just the cherry on top, and this article is intended to be about Brave Software and the Brave Browser mostly.
Less than a year later, on January 20th 2016, they launched the first version of the Brave Browser. They integrated something very similar to the often ignored "Acceptable Ads" policy - allowing trackerless ads by default, and outlining plans to insert their own trackerless ads into webpages. 
Two years later, in June 2018, a pay-to-surf test version of the Brave browser was released. It was pre-loaded with 250 ads, and sent a detailed log of the user's browsing activity to Brave, although they pinky-promised not to share it and only use it for research for their "machine learning"-based ad insertion system. 
Oh yeah, they were using a supposedly hardened Electron fork called "Muon" until December 2018, then they switched to just being a Chromium fork. 
Brave Bravely Rips Off Tom Scott
Shortly before Christmas in 2018, Tom Scott, a youtuber who makes rather interesting and educational videos, was asked if any of the money being donated through Brave's "BAT" system (some cryptocurrency blockchain BS), he realized that the browser was fooling people into thinking they were supporting him, when in reality he didn't see a single penny. Eventually, this whole fiasco ended with Brave adding a warning to "unverified" creators, and with Tom sending Brave Software a formal GDPR "right to be forgotten" request. 
Refer Me Back To Firefox, Please
In June 2020, twitter user @cryptonator1337 noted that Brave was automatically injecting referral codes into URLs for the cryptocurrency exchange site Binance. If you typed "binance.us" into the URL bar and pressed enter, Brave would instead take you to "binance.us/?ref=35089877".
Brave disabled this in the code several hours later, in a typical "we're sorry we got caught" fashion, and a Brave employee saying it was simply a "clean way to support brave".
A proper way to do this would have been simply a one-time popup when visiting this site, asking if you were okay with it or not. Most of the time, explicitly opt-in things are perfectly okay.
And Another Thing
It's simple. In 2008, he had donated $1,000 to support passing California Proposition 8 (which was a ban on same-sex marriage), and $2,100 to Tom McClintock, a Prop 8 supporter. It's no secret that Mozilla is a progressive company (a good thing), and as a result, half of Mozilla's board stepped down .
Eich publically apologized and promised to help LGBTQ+ people, but he still resigned and severed ties from Mozilla on April 3rd, 2014.
But what about...
Many people still prefer to use Brave over a more private browser such as Firefox for numerous reasons, many of which are based on flawed reasoning.
Brave has a built-in adblocker!
uBlock Origin is one of the best and most respectable adblocking extensions currently, and it works best on Firefox.
It's best for extensions to remain separate from the core browser anyhow, as they often do not follow each other's update cycles (imagine needing to update your entire browser to fix a bug in the adblocker!)
Chromium, and therefore Brave, is more secure than Firefox.
It is true that Chromium already has stabilized process isolation (separaing different tabs into different processes), but Firefox is in the process of doing so too, via Project Fission. In fact, you can enable it now via
fission.autostart in Firefox Nightly!
There is not much that can be done for security (which is separate from privacy) other than process isolation, ASLR, or DEP. Most security issues are, of course, as a result of exploits, and whaddya know, Mozilla has a programming language intended to help prevent entire classes of bugs (memory unsafety, data races, etc). Sure, Rust isn't a magic bullet, but to say that it's nigh-useless for preventing vulnerabilities is not only wrong, it is utterly stupid.
But My Benchmarks!
Inevitably, someone will mention benchmarks if the discussion is about competing browsers. Guess what? Synthetic benchmarks are not representative of user experience, and most people will notice little to no speed difference between modern browsers.
Userbenchmark, while they don't benchmark browsers, are a really good example of why benchmarks are a dumb way to measure whether one thing is better than another. According to them, an i9-9900K is 7% better than a Ryzen 9 5900X, even though the supposed "differences" they mention would never be noticed by even a power user (In addition, it's quite well-known that UBM is biased towards Intel and Nvidia)
Benchmarks are funky. They're helpful for developers. They shouldn't mean jack-shit to an end-user in most cases.
But I don't want to use Firefox...
Fine. Use Ungoogled Chromium.
It's stock Chromium with a deep "de-Googling" script applied; removing any and all integration with Google services, telemetry, etc.
WARNING: Do be aware that Ungoogled Chromium is a volunteer project, be wary when using pre-compiled binaries. They're probably safe, but no guarantees. If you're on Arch Linux, doing
paru -S ungoogled-chromium will compile it yourself.
 Lomas, Natasha. "Blockchain Browser Brave Starts Opt-in Testing of on-Device Ad Targeting." TechCrunch, 20 June 2018, techcrunch.com/2018/06/20/blockchain-browser-brave-starts-opt-in-testing-of-on-device-ad-targeting. (archive.org)
 Cimpanu, Catalin. "Brave Browser Moves to Chromium Codebase, Now Supports Chrome Extensions." ZDNet, 13 Dec. 2018, www.zdnet.com/article/brave-browser-moves-to-chromium-codebase-now-supports-chrome-extensions. (archive.org)
 Gerard, David. "Brave Web Browser No Longer Claims to Fundraise on Behalf of Others — so That’s Nice." Attack of the 50 Foot Blockchain, 7 June 2020, davidgerard.co.uk/blockchain/2019/01/13/brave-web-browser-no-longer-claims-to-fundraise-on-behalf-of-others-so-thats-nice. (archive.org)
 Castor, Amy. "Brave Browser Is Collecting Donations on Your Behalf — Did You Know?" The Block, 24 Dec. 2018, www.theblockcrypto.com/2018/12/24/brave-browser-is-collecting-donations-on-your-behalf-did-you-know. (link is already an archive)
 Machkovech, Sam. "New Mozilla CEO Issues Statement, Expresses ‘Sorrow for Causing Pain.’" Ars Technica, 26 Mar. 2014, arstechnica.com/information-technology/2014/03/new-mozilla-ceo-issues-statement-expresses-sorrow-for-causing-pain. (archive.org)
 Shankland, Stephen. "Mozilla under Fire: Inside the 9-Day Reign of Fallen CEO Brendan Eich." CNET, 13 June 2014, www.cnet.com/news/mozilla-under-fire-inside-the-9-day-reign-of-fallen-ceo-brendan-eich. (archive.org)
 Newton, Casey. "Outfoxed: How Protests Forced Mozilla's CEO to Resign in 11 Days." The Verge, 3 Apr. 2014, www.theverge.com/2014/4/3/5579516/outfoxed-how-protests-forced-mozillas-ceo-to-resign-in-11-days. (archive.org)