Be aware my edits is in conformity with OSM rule. https://wiki.openstreetmap.org/wiki/Disputes#On_the_Ground_Rule
Almost roads in Japan don't have highway name. It is just operator and reference number. 東海道 is not currently road name but used in places because it is popular.
If we follow our policy, we need input bridge or tunnel name to name key. If we follow your opinion, we need input bridge or tunnel name to name key too because national routes use them as highway name.
By the way, what source of your this opinion? "There's nothing about which name is more prominently shown on signs." Almost known national routes in Japan are differ your opinion.