How can I require approval of any post before it's published?and my typical answer will be
You can't, using the post editor. Any author can publish a post, and you can only moderate posts after the fact.
The key words there are "post editor".
If you enable Mail-to-Blogger in Settings - Email, you can have your authors email their posts. If you then set Mail-to-Blogger to save emails as draft posts, all posts emailed will be saved for you to moderate. In Settings - Mobile and email - Posting using email, select
Save emails as draft post
You, or any administrator, can examine draft posts from the Edit Posts menu, and publish only the ones that meet your approval.
And going better, if you use a GMail account to filter and forward to MTB, you can selectively route posts from specific individuals for moderating, and others can post directly. This allows you to use your time more effectively, only moderate possible problem authors, moderate more effectively, and respond more promptly. If you can respond more promptly, you can selectively moderate based upon need, not just current workload.
You can change individual author moderation policy on a moments notice. Setup 2 Blogger accounts for your blog, each with a different MTB policy (Account A to post after moderation, Account B to post without moderation). Have your authors email their posts to a GMail account.
Setup the GMail account with a series of email filters, with each author email forwarding to the appropriate Blogger account MTB email address. Any individual author can be selectively changed between moderation and no moderation, by simply changing the email filter to redirect to Account A or Account B, respectively. You spend your time moderating only authors posting through Account A.
If you want to publish the posts under the individual author names and profiles, setup individual Blogger accounts, as blog authors - one for each author - and only give the authors the Mail-to-Blogger email addresses. Then, as above, setup each Blogger account to save emails as draft posts. Since you will be the actual owner of each Blogger account, setup the accounts with secure names and passwords - and keep track of the account names and passwords.