You can approve media that's pending approval from backend within CB Gallery > Media. Using the Search Tools at the top of the table you can filter it to Unapproved media as well. You can also approve media from frontend by creating a Joomla menu item with a menu type of Community Builder > Plugin then for Plugin select CB Gallery and finally set Action to Approval. Then of course as you're already doing you can approve the media from the gallery it was uploaded to.

OK, thanks. I see that works if the user uploads a file - as long as they don't add it to an album! Not sure if that's a "hidden configuration option", but they can still upload any file to an album without required approval.

Album uploads can not bypass approval. Approval doesn't care that it's being uploaded to an album or not. The only uploads that skip approval are those directly uploaded from a stream through CB Activity as long as the type isn't a file. CB Moderators are also completely exempt from upload approval and their uploads will be published immediately so be sure Moderator View Access Level is configured correctly in CB > Moderations.

But I needed an easier method for my non-technical users to submit content. The Joomla create article function is too difficult.

Not sure how others handle this but I found I could just give them a Submit Content button using the BreezingForms extension which gives them a page to upload any file ( extension & size restrictions apply) to the admin editors.

That doesn't really have anything to do with CB so I can't really help you there. I've always just used Joomlas create article page, but yes you could use a 3rd party form extension or look into using something like K2.