I would assume that once a lot of advertisers are on the platform the ad catalogue can quickly grow to the size of GBs, which would be unfeasible to hold on smartphones.
(I work at Brave, and am on the team bringing the ad platform to market)
Also, what are edge cached urls?
If you click the notification, Brave opens a tab in the Brave browser and navigates to the advertisement location. At this point, the ad can load images, video, etc.
Important to note, however, that ad pages are not given any special treatment in the Brave browser. They are subject to the same privacy/security restrictions on all other pages. Third parties are severely limited, if not entirely prevented from engaging in the session.