There was a change in the meaning of highway=path a few years ago.
Now it is better to treat it is as just some vague way. The only certain thing is that it is not suitable for motorcars. 
highway=footway has rather strict meaning "only for pedestrians", so mapper should at least remember about cyclist and add bicycle=yes, if it makes sense of course.
Here we have connection between combined footway/cycleway and a public road. 
Thus this connection should allow bicycle routing. Change to highway=path is the simplest method.