I can confirm this issue, or is it a different issue? I just encountered it on MacOS 10.13.14 and Brave 0.22.22 version.
There are two big usability problems in Brave, after all open windows have been closed.
Clicking on the Brave icon, if the user has the icon in the Dock (app launcher at bottom center of screen), does almost nothing visually, except switch the focus of the menu bar to Brave (this is subtle, and can be hard to notice at times). It casually looks to the user like Brave has crashed or gotten stuck, because the user was expecting a new window to appear. The expected behavior is to open a new window, as if the Brave program had been cleanly closed and then freshly opened again. Google Chrome already does this, on MacOS.
Clicking on a hyperlink from another program, when Brave is the default browser, does nothing at all visually. Although Brave has received the message, there is no window open, so Brave does nothing with the URL passed to it. To the user, it looks like the hyperlink wasn’t clicked on properly or the program is not responding, so the user typically tries clicking again. Again, nothing happens.
Interestingly, Brave still is alive, and if you switch focus to the Brave menu bar (this can be done by clicking on the Brave icon in the Dock), you can use the File menu, in the menu bar, to give the “New Window” command, and then it will work! A new Brave window appears, and interestingly enough, in it will be all the hyperlinks the user has clicked on in step 2 above, usually multiple copies of them, because the user has clicked several times in frustration.
MacOS is somewhat unique among modern operating systems, in that it allows a process to continue to exist, even after the final window has been closed. Closing the final window does not close the process, as you might think. Since Brave is cross-platform, it might have been coded to assume this, and so it breaks on MacOS in this way.
Is there already a bug in GitHub for this behavior, or is this a different bug?