I'm a broken record at this point, but Price's issue being that he's worked himself into a weapon. Actually, scratch that. A weapon has no autonomy, and Price surely calls the shots. He's war. He's the undertaker. He'll never be as good at life as he is at escorting death to people's beds and doorsteps. All he will ever use his hands and brain for is destruction. Until a wrong step, a bad piece of intel, a stray bullet, finally takes him out.
But he's not a psychopath. Conditioning, trauma and sheer will have freed him from the levels of guilt and empathy the average man is burdened with, but he's not truly a maniac. And more is the pity; a certified psycho wouldn't have to deal with the pesky fears and desires that come to the surface from time to time. More so as he gets older.
And Kyle's problem being that he's so angry — so young, so driven, and so, so weak and powerless under the weight of his own boundless fury. He needs a target. And, sure, war gives him plenty of that, specially with Price calling the shots. So Gaz lets Price take the wheel, in every aspect. If Price says the world is full of plotting villains, of scheming shadows, terrorists left and right who need elimination (and some of them might look quite innocent, some might be unarmed. But that's okay, just trust your Captain. He knows everything), that's more than fine by Gaz. More nails for his hammer.
But his anger is of a deeper, more intimate nature. It comes from the home, from inside his veins. It can't be abated by your average faceless target, not really. His mortal enemy (the father, the introduction to the closed fist, the first betrayal) is a silhouette that, under the right conditions, coalesces into a shape not too dissimilar from his Captain's.
So when the boss, the man steering the wheel, the grim reaper who presses crimson dots between people's eyes and whispers, "there," shows little silvery cracks, when his gravely voice weavers, when Gaz talks about leaving and Price's impassive face twitches and Gaz (taught by the best) steals proofs of weakness, of the lonely old man within... It's like a shark tasting blood in the water.
Price is a ductile thing — resilient, though not so hard as to break. He's a difficult man to humble, to corner. But now Gaz knows he's not the only one who needs something. He's not the only human in this equation, with squishy, squashable insides. Price is as afraid of sitting alone in this hell of his own making, as Gaz is afraid of not being allowed to enter.
And how can Gaz exploit this intel, this leverage? Blackmail? Manipulate Price to progress his career? Nothing as pedestrian as that. He just needs to hear Price admit a thing or two. Maybe make him beg a little.