Being a Warden allowed you to control armies. But, that power was contingent on the Seven Kingdoms being intact. The outbreak of the War of the 5 Kings fractured the Seven Kingdoms and seemingly left a Warden (especially in the North) with no resources to control.
But, in Season 5, episode 6, Baelish had this dialogue with Cersei:
BAELISH: Stannis Baratheon is also a traitor, marching with his army on Winterfell. Let Stannis and Roose battle, let the enemies of the throne slaughter each other and when they’re done seize Winterfell from whichever thief survives.
CERSEI: Winterfell is 1,000 miles away from here. The weather has already begun to turn.
BAELISH: That is why it is critical to strike soon, while the victor still licks his wounds. Perhaps I can help. The Knights of the Vale are some of the best fighters in Westeros, trained to battle in the ice and the snow.
CERSEI: And if you succeed?
BAELISH: Name me Warden of the North.
After Battle of Bastards, Baelish could have easily done what he said he would do. But, instead, he had his forces stand down.
A King in the North would control all the forces the Warden of the North once controlled?
Baelish knew that the North basically already seceded with Robb Stark declared as King in the North, yet Baelish still wanted the title of Warden of the North?
A final twist, in season 6, episode 9, is that Baelish told Sansa that the Vale has declared for the King in the North, thus betraying the Iron Throne, right? Or, maybe he lied and betrayed Sansa?