It doesn't update automatically because brave is not in the official repository.
You don't have to download and reinstall.
You can add the brave ppa repository for your Ubuntu version to your apt repository file ( there are hundreds or thousands of howtos on this already, so no need to rewrite here.)
Or you can use the brave snap version.
Or appimage or flat pack if they offer it.
But basically Ubuntu doesn't update it because it isn't Ubuntu software. Lots of other browsers are, and resources are limited.