Having worked in marketing, they absolutely worked with a data broker. You don't blindly send out mailers. You use data brokers to zone in on a target market. "This zip code is more like to buy X while this zip code is more like to buy Y."
All direct mail marketers do this, because it allows better targeting (& re-targeting old prospects & previous customers) but Brave explicitly said they didn't, and only went through the USPS's service.
The USPS didn’t mess up. They are clearly using some sort of vendor to send out the mail.