Pretty house. At this point, if it were my house, I would cut a 2-foot-square section of concrete, dig a 3-foot-deep hole, fill it with rocks or some sort of dry well, put a pipe for a downspout, patch the concrete, and connect a downspout to the pipe. The section of roof draining to that gutter is very small, so I don't think that such a dry well would ever overfill.
If someone just hadn't installed the gutter right there this would have never been noticed and no one would have given it a second thought. This is all pretty silly. If ice becomes an issue - Watch Your Step!