Is there any reason this needs to be on a public blockchain? What are the benefits? Why is this a better solution than Brave keeping points/payments data in their own database?
The use of crypto here has the same reason as use of crypto almost anywhere else, to sidestep regulations and liabilities. Doesn't matter whether those regulations are there for very good reason.
That might work in a jurisdiction where the trade in cryptocurrency is illegal but its possession is allowed, but I can't imagine a US or European court would take that argument seriously when crypto has a demonstrable market value. The court would almost certainly treat crypto like a commodity in that case. Wheat doesn't have KYC requirements, but if you knowingly and demonstrably facilitate wheat trades on behalf of a sanctioned entity, you can expect some legal trouble.
Cf. e.g., from 2018, https://thenextweb.com/news/brave-blockchain-cryptocurrency-...
Blockchains are interesting, they're useful sometimes.
1. Filecoin: It was well-motivated from the start, to provide a cheap way to get counterparties (by default, a multiple of six) to pin content in IPFS. I don't regard it as a proven technology yet, but it's usable as it stands;
2. Brave's BATs: Brave annoys me with its cryptocoin boosterism, but in principle this provides an way for content providers to profit from gaining a readership that doesn't involve nasty ad-tech. I don't think the system provides good enough incentives for users to mint BATS as it stands, but I think that in principle the Brave team have a viable solution to this problem.
3. The Stellar payments infrastructure has a cryptocoin used to grease its wheels. I don't see Stellar's choice to mint their own cryptocoin as principled here (they could have allowed nodes to choose), but using some cryptocoin here is driven by use.
None of these coins use Bitcoin / present-day Ethereum style proof of work, so they do not appear to have the negative externalities you deplore.