Why does a Thrust Vector Control System increase maneuverability?
The point of thrust vector control is post-stall maneuvering.
Control surfaces lose much of their effectiveness when the flow over them is separated. Pointing the jet exhaust in different directions works almost regardless of angle of attack. All that is required is good intake flow, and by placing the intake below the forebody (as done on the X-31, the F-16 or the EF-2000), this is still possible even at high angles of attack.
Post-stall maneuvering enables the airplane to point its sensors and weapons in the direction of an opponent within seconds of detection. Compare that to flying a turn at Mach 0.8 over maybe 150°. At 6g (= 80° bank angle), this takes 11.5 seconds, assuming a speed of 256 m/s (which is Mach 0.8 at medium altitude) and an instantly achieved maximum turn rate.
By attaining a very high angle of attack in order to slow down the airplane and thrust vector controlled rotation of the fuselage into the direction of the opponent (Herbst maneuver), an airplane can turn much quicker and, consequently, shoot earlier. This is critical for winning in air-to-air combat.