This isn’t a typical support request because I have fixed the problem, but it was a bit difficult for a non-technical user, so it might give you a useful heads-up.
A few minutes ago I noticed on About Brave that my version was at 1.15 whereas the latest is 1.29. Update failed with code 80072F7D. I found this ticket which states that you can fix it with the offline installer.
However it was really difficult to find an offline installer because it seems that you guys are pushing for the downloader setup instead, even on external sites such as Techspot and Filehippo.
However, the downloader setup wouldn’t update Brave either. Firewall was disabled.
In the end I found the offline installer at Softpedia which fixed the problem and now I’m in the latest 1.29 version.
So, until you guys fix this bug, please make sure to provide an easily accessible offline option.