Why do all Japanese armor have headdresses?

The headdress of the ancient Japanese armor was called "tateboshi", and not all helmets had it, it was popular from the Heian period until the end of the Shogunate:

Most of the countries in the world put decorations on the top of their helmets, but since the top of the helmet of the Japanese helmets was a tenshin (heavenly side) seat, there was no way to put decorations, so they came up with a new way to do it. However, since the top of the Japanese helmet is a sky seat, there is no way to place the ornament, so they came up with the idea of placing the ornament on the front of the helmet. And because the decorations in the front of the helmet, in order to let the enemy and the two sides can see clearly, so deliberately made high and large, and finally formed the unique Japanese helmet in this spade shape.

The origin of this decoration is actually from the ornamentation of the helmet of the Tang Dynasty: