Since the force that acts on a bullet when a gun is fired is equal and opposite to the force that acts on the gun, doesn't this imply a zero net force, and therefore the impossibility of an accelerating bullet?

The gun and bullet both will have equal and opposite force on it and it will accelerate the gun and bullet in opposite directions

Explanation:

When bullet and gun system is considered then the force due to spring on the bullet is the accelerating force on the bullet.

Now by the Newton's III law we know that force due to gun on bullet is always equal in magnitude and opposite in direction to the force due to bullet on the gun.

So here if we consider both as a system then the two forces will cancel each other and we can say that net force on the system of Gun + Bullet is zero

So here the acceleration of center of mass of gun + bullet system is zero

but if we consider on bullet then the force due to spring on the bullet is non zero and unbalanced force.

So the bullet will accelerate and reach to high speed at the end