I agree that semantically the word "construction" is not ideal but it is used to denote road "impassable by the traffic it is intended for", and they even give example cases which seems to fit the situation:https://wiki.openstreetmap.org/wiki/Tag:highway%3Dconstruction#How_to_map_the_start_of_a_road_repair_or_reconstruction

About the routing engines, it sounds infeasible to parse access:contionnal value, since it's an open field with no rule to actually implement. I guess that's why the construction value exist for highway tag, to have a close set of value for routing engines to take decision