While in theory this idea could benifit a significant number of people, in practice this would be a useless construction job. Sure, it would help some middle class travelers travel across continents for less money, but would be extremely time consuming and in fact quite dangerous in the winter. In addition to this, the currents in the Arctic sea are quite strong and often dangerous, especially in the winter. The upkeep would be incredibly expensive and the road would be unsafe to travelers in winter.
Admittedly, a rail line could possibly be a useful trade outlet, but a road simply would not get the job done.