According to Newton's 3rd law, the force upon the shootee must equal to the force upon the shooter.
Each force has an equal an opposing force, just as it should, just as Newton's law says it does, but that doesn't mean that all the forces in a problem are equal.
It's not possible to chain together a bunch of forces and say they are all equal just because they are all generally linked together by a common problem.
For example, the force on the rifle and the bullet are both due to the pressure of the discharge. They are equal.
The force of the bullet on the target and the target on the bullet are equal, but because they are both dependent on how fast the bullet is decelerated by the target and not on how fast the bullet is accelerated by the discharge pressure, they are not equal to the force on the bullet and the rifle created by the discharge pressure.
The force of the stock on the shoulder and the shoulder on the stock are equal, but because they are dependent on how fast the rifle is decelerated by the shoulder, not on how fast the bullet is accelerated by the discharge pressure or on how fast the bullet is decelerated by the target medium, they are not equal to the force of the bullet on the target nor to the force on the bullet created by the discharge pressure.
Force on the shoulder by the stock = Force on the stock by the shoulder.
Which does not equal...
Force on the rifle created by the discharge = Force on the bullet from firing.
Which does not equal...
Force on the target by the bullet impact = Force on the bullet by the target impact.
Which does not equal...
Force on the shoulder by the stock = Force on the stock by the shoulder.
But if we be blindly apply the law, it may not be right.
If we apply the law incorrectly then the results are nonsensical. That doesn't mean the law is wrong or invalid in some scenarios, it just means that it needs to be applied correctly to provide valid results.
For example, the Pythagorean Theorem says that it is possible to calculate the hypotenuse of a right triangle by a certain formula. If the formula is used to calculate the side of a triangle that is not a right triangle, it won't work. Not because the formula is wrong, but because it was applied incorrectly.
Newton's law does not say that all the forces in a given physics problem are equal, it only links certain forces together. Attempting to link all of the forces in a problem together and set them all equal to each other will lead to misunderstandings, but that is not a problem with Newton's law, it is a problem with the way it is being applied.
Newton's law says that for every force there is an equal and opposite force. In this problem, we see that we can group all the forces into pairs that are equal. Which is exactly what Newton's law says should happen. Newton's law does not guarantee universal equality for all the forces in a scenario, it only guarantees that the forces in a scenario will occur in equal pairs acting in opposite directions.
While we are at it, is it physically possible to shoot a man flying off his feet? Physically means according to physics.
There are two problems with shooting a person off their feet. First of all, humans have a lot of "give" to them. Even if the impact of the bullet imparts tremendous force, a lot of that force will be used up deforming and destroying tissue (and the bullet as well), and therefore won't be available to actually move the person around significantly.
Second, humans act a lot like water and impacts to water will apply force in all directions, not just straight forward. When a pebble is dropped into a pond, the ripples go out in all directions, and that means force had to have been applied in all directions to make those ripples. The same thing happens to the human body.
All of that means that even though tremendous force is applied by the impact, it won't move the person around much. It will poke a hole and stretch/tear/deform things, but it's not going to move the entire person very much at all in the direction of bullet travel.
We can see this is true when shooting small varmints with large caliber rifles. It doesn't send them flying backwards nearly as much as it just splatters them around in all directions. If there is a prevailing direction to the movement of the largest part of the carcass, it tends to be the result of jet effects from the fluid being ejected by the impact more than the result of the direction of the bullet travel.