I think the road classification is correct, unclassified, but I'm concerned about the access=private. Are there signs posted to limit access to the road? If so, then access=private would be correct. 

Do you mean to show the road is privately owned and maintained? Then use operator= or ownership=private. Let me know if you have any questions or need help.
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            Published using OSMCha: https://osmcha.org/changesets/146158027
            