Just wanted to let you know we're looking in to this. The documentation definitely makes it seem like you can calculate shipping rates using this endpoint, but to be honest I don't remember ever seeing that and I'm not seeing a way to do it now. I'll post back here when I have an update for you.
Just want to throw my two cents in here! I've also been trying to get this to work based on what I read in the docs, but I can't figure it out. Seems like I'm not alone! Any word on a possible fix?
This is an accepted solution.
I did find a way to get this working. Take a look at the Checkouts API. There is an endpoint to shipping_rates.json that returns the shipping rates. All you need is the checkout token. I was able to get this from the draft order's invoice URL. For example, draft order invoice URL: https://apple.myshopify.com/690933842/invoices/d64d93d528cfc75c0876183d58f4a has checkout token: d64d93d528cfc75c0876183d58f4a.