We are in process of updating the documentation because you're right that it needs more work. For the record, if you're doing everything on your own via Cost Explorer, Storage Lens and processing CUR you may be set. From what we hear, most folks do not want to deal with processing CUR (or even know what it is) and struggle with Cost Explorer.
Vantage automates everything you just mentioned to allow you to make quicker decisions. Here's a screenshot of what we do on a S3 Bucket basis: https://s3.amazonaws.com/assets.vantage.sh/www/s3_example.pn...
We'll profile storage classes and the number of objects, tell you the exact cost of turning on things like intelligent tiering and how much that will cost with specific potential savings. This is all done out of the box, automatically - and we profile List/Describe APIs to always discover newly created S3 Buckets.
From speaking with hundreds of customers, I can also assure you that at a certain scale, billing does not take an hour...there are entire teams built around this at larger companies.