To answer my own question:
The example policy granted PutObject access, but I also had to grant PutObjectAcl access.
I had to change
"s3:PutObject",
"s3:GetObject",
"s3:DeleteObject"
from the example to:
"s3:PutObject",
"s3:PutObjectAcl",
"s3:GetObject",
"s3:GetObjectAcl",
"s3:DeleteObject"
You also need to make sure your bucket is configured for clients to set a public-accessible ACL by unticking these two boxes:
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