AmazonS3FullAccess managed policy on a group doesn't give S3 permission?

I have an S3 bucket that has in its policy permission for my CloudFront origin access identity:

{
  "Version": "2008-10-17",
  "Id": "PolicyForCloudFrontPrivateContent",
  "Statement": [
    {
      "Sid": "1",
      "Effect": "Allow",
      "Principal": {
        "AWS": "arn:aws:iam::cloudfront:user/CloudFront Origin Access Identity <mine>"
      },
      "Action": "s3:GetObject",
      "Resource": "arn:aws:s3:::<my-bucket>/*"
    }
  ]
}

Additionally I've created a group and attached the AmazonS3FullAccess managed policy to it and added an IAM user to that group. The managed policy looks like this:

{
  "Version": "2012-10-17",
  "Statement": [
    {
      "Effect": "Allow",
      "Action": "s3:*",
      "Resource": "*"
    }
  ]
}

However when that user tries to add anything to the bucket, I get a 403 (access denied). I'm unsure if any other operations work, I haven't written code to try them. It's only when I specifically allow that user's ARN access to the bucket directly in the bucket policy that they're allowed to add objects. What am I missing? It seems like the above group policy should allow members of that group access to all operations in all buckets, but it doesn't do that.