Several people pointed out a good point here , Why did Ulfric had to use a shout ..should make people wonder, not me i already know Ulfric is on a personal powertrip
I believe he was trying to show the Nords that they have power in their ancient traditions that they aren't using, and that relying on the legions has made them weak.
It helps to understand Germanic and Norse culture. Germanic kingship was not automatically handed down to the king's son, and these cultures had no "divine right of kings" prior to taking that idea over from the Romans. The king was chosen by the
thing or folksmoot based on his war prowess and courage. If he couldn't fight, he wasn't able to defend his people. Having the strongest man in charge means survival.
Skyrim is under an existential threat from the Thalmor, so having a weak, indecisive king is a risk they cannot afford.