Sounds like you’re doing everything right.
When you import the files to the new site, does it tell you the import was successful?
You should see your imported settings in the Customizer immediately after importing.
I just did this to another site the other day, so I would assume there is some sort of permissions issue.
Who are you hosted with? Have you contacted them to ask if there’s anything that could be preventing the importing of the json file?
Let me know 🙂