Do I need (will I ever need) Composer or Bee (or anything else)...

No. There are no plans to make Composer mandatory and most people don't touch it for Backdrop sites.  While there are a few things you can do in Bee that you can't do in the UI, you can easily manage your site updates almost entirely through the user interface and without Bee.  

One exception, but you can do this by editing the files through cPanel File Manager, FTP or some other way is when robots.txt, .htaccess or settings.php have changes that you need to apply.  These are not replaced in the UI upgrade process (and won't be when the Bee upgrade command is complete) as it could override any custom settings.