Hi again,

For this situation located here and my first-hand experience on the ground, 'access=permissive' or 'access=destination' are the most appropriate. I will share my rationale of how I came to the conclusion to use access=destination.

From the Wikis:

Note that access=private is intended to indicate that access is restricted, not whether the object is privately owned or not. Use ownership=private or operator:type=private to record this kind of status. For example, a privately owned road with public access may be tagged like any other road with public access – without access=* tag, or with the explicit access=permissive.

Unfortunately as of 2023 access=private is widely used as a duplicate of access=permissive as well as ownership=private and should not be expected to match the originally intended meaning.

However, since access=permissive indicates that access can be revoked, (Ie, a gate that is commonly left open) Permissive doesn't seem applicable since there is nothing to physically restrict access located here. 

Access=destination seems the most fitting since the intended use is to stop navigation apps as you desire: ""no thru traffic" / "local traffic only"