@Elsie
The Blue Cloud symbol

means that the app has previously been installed on at least one of your iOS devices, and can be downloaded again.
Any chance that you have previously enabled automatic offloading of unused iOS apps in Settings? Or, the iOS update changed settings on your iPhone - where you would not have expected?
Every iOS update I have ever run, has been followed by my having to go over several settings in order to locate some that were changed by the iOS update.
Please double-check something:
On your iPhone, please go to: Settings > ScreenTime > Content & Privacy Restrictions
If Content & Privacy Restrictions is Enabled, then see: Settings > ScreenTime > Content & Privacy Restrictions > iTunes & App Store Purchases ← therein, set:
Installing Apps to Allow
- and for
REQUIRE PASSWORD set to Don't Require
And then back your way out of the Settings.
The Files.app (which is yet another iOS application like the Mail.app, and another way of writing Files app) is also shown in the Apple App Store, simply as Files (You can search for that, in the App Store. And, you can look up online, for tips on how to use the Files app.).
The Brave folder within the Files directory (as I have previously written about) on your iPhone, would be one location where you could store backed up / ie exported Brave Browser bookmarks files (exported, these files will be named bookmarks.html).
The general idea, behind my posts, is to be sure that you are prepared - “just in case” - for either:
- the event of a thorough un-install (that includes deleting Brave data) and then re-install of Brave Browser (iOS)
- an install or re-install of Brave Browser (iOS)
I do not know the exact Apple iOS activity that occurs during a complete, “forced” / “hard” Shutdown of the iPhone, except I do know that action has many times - somehow - fixed issues with the iPhone or some software misbehaving.
Occasionally, I have had to do the complete Shut Down after a thorough removal of Brave Browser, in order to thereafter do a fresh installation of Brave Browser . . . that works well.
On such occasions, oddball Brave Browser performance was the issue and motivating factor. On one occasion, a particular Brave Browser setting that was supposed to exist in Brave Browser (iOS), did not exist.
I do not know why; it just plain did not exist. Then, going to the trouble of backing up my Brave Browser data, followed by a thorough removal of Brave Browser, followed by a complete shutdown of my iPhone, followed by a fresh installation of Brave Browser . . . worked.