A road cannot move; the road isn't going home (as in "the road is moving to the house itself"). The subject for "가-" is omitted. Here, "집으로 가는 길" literally translates to "the road along which one goes home".
Another example of the need for context. You could compose a sentence where 지브로 간 길 would indeed translate "the road home." He chose the wrong road home. 집으로 간 길 잘못 걸렸다. The road home was the last road he ever traveled. 집으로 간 길은 마지막이였다.