On a continent. City on coast. A one-hex bay - mouth of a river - separated the resource hex from the city. The City, which had a trade route, had expanded across the bay but only to the resource tile. When the resource tile was improved, my civ had no credit for it. What I finally did was to construct a road to the resource hex upsteam across the river. Then, I got credit for having the resource.
This was in an early release some time last fall. I have had no problem with resources on islands . Only with this situation.
If I had waited and not built the road, I would not have been able to get access to the resource until I had culture growth which connected the hex by land. Once this happened, I removed the road.
If I absolutely had no reason to build the road in that peculiar situation, I wouldn't have built it. But actual reality trumps rules and what "should" happen. This may have been fixed by now. It shouldn't have happened then, so maybe it has been fixed.
So, I would say you shouldn't have to build roads to connect resources, but if you don't get credit for having an improved resource within your cultural borders because of the lay of the land and you need its benefits, try building a road. You can always delete it later.