Right. Momentum is the quantity that is equal and opposite in a gun, so it is what conveys force and resulting velocity in recoil, not energy.
The classic firearms example is also in studying recoil. Many if not most folks have the idea a gun doesn't injure the shooter as much as the
shootee because the contact area of the gun is bigger for the shooter. But that area difference is typically small compared to the difference in kinetic energy in the bullet and in the recoiling gun.
Example:
A 150 grain bullet is fired from a 8.5 lb rifle at 2500 ft/s. The rifle is 397 times heavier than the bullet, so, since they both have the same momentum in opposite directions, the final velocity of the rifle 397 times smaller than the bullet velocity, or 6.3 ft/s. When you calculate the kinetic energy, you first convert the weights of the bullet and gun to slugs of mass:
Grains/225218=slugs¹, and pounds/32.17405=slugs, so
150 grn/225218 grn/slug =0.00066602 slugs
8.5 lb/32.174 lb/slug = 0.2642 slugs
KE = ½mV²
KE of bullet = 0.5 × 0.0006602 × 2500 × 2500 = 2081 ft-lb
KE of rifle recoil = 0.5 × 0.2642 × 6.3 × 6.3 = 5.24 ft-lb
5.24/2081=0.00252
So, the shooter gets hit with 0.00252 times the energy the shootee does. Additionally, if you suppose the contact area of the butt of the rifle is 2 square inches and the bullet cross-section is .0745 in², there is about 27 times more area on the shooter absorbing the blow, but it is that huge difference in KE, the measure of work the recoiling gun can do on the shooter, that accounts for most of the shooter's relative comfort. So, the fact the gun weighs more than the bullet is the source of protection.
In 1720 Willem 's Gravesande dropped different brass balls into clay and discovered that when he doubled the drop hight the ball penetrated 4 times as far. As s student of Newton's work, he realized that because gravity is an accelerating frame of reference, the balls dropped from twice the height were going twice as fast. But since they penetrated four times as far, the penetration of clay was therefore proportional to the square of velocity. Kinetic energy is also proportional to the square of velocity, and even though it wasn't defined clearly until almost a century later, this is the historical root of its role in terminal ballistics.
Of course, 's Gravesande was working with relatively slow velocities. When energy is conveyed at high velocities, because it is a scalar quantity not tied to any particular direction as a vector quantity like momentum is, the work can go not only into penetration but into spreading a shock wave transversely and that confuses its role and merit in terminal ballistics. But overall, if two bullets of the same type and construction have the same kinetic energy, the one that has the greater momentum will penetrate ballistic gelatin further. There are a lot of examples of this comparing 9 mm and 45 Auto. But get into penetrating hard barriers and the roles can reverse, the difference being in how fast energy must convert to work to break through them.
¹
Pounds × 7000 equals grains, and 7000 × 32.174 = 225218