Most of the permissions are required for Brief to work, unfortunately. The ones that could be optional are bookmark access (used because items starred in Brief are the ones bookmarked in Firefox, which could be changed), notifications (feed update notifications are enabled by default), downloads access (optional for now, but will be needed to implement automatic backup). All the worst ones - access data for all sites, access tabs, store unlimited data - are the ones required for a feed reader with long-term storage of the feed entries to work in Firefox.
If you have ideas how to make Brief work without requesting these, would be cool to hear them on the Brief's GitHub issue tracker.