Roads do not need access tags to be routable. It is assumed all roads are public, unless specified otherwise (using an access= tag). The value of access tags should also not be assumed - =destination is only appropriate when there's a sign at that road indicating such (for example one that says "no thru traffic").
I encourage you to read through this article, if you haven't already: https://wiki.openstreetmap.org/wiki/Key:access