When we calculate things for a system that is in its most disorganised state, the best approach is to aim for average values. Have some patience and you'll know why.

Backstory

Let's consider a container. Don't worry about the shape, it won't matter. There is an ideal gas filled in it. The container is sealed to its best to avoid any leakages.

Averaging the stuffs

Now consider a molecule of the gas inside it. It has a randomly directed velocity. Now consider all the molecules of the gas. Their speeds might be same or different depending on multiple factors, though variance in speed will be limited and almost all the molecules will be moving in a fixed range os speeds. But their velocities remain randomly directed. And so will their momenta. Any randomly directed vector of magnitude in small variation from the mean value averaged over a very large number of observations is zero. So is their velocities and hence net momentum of the system. So if one molecule hits the wall of container in a direction, in the opposite direction(s), some other molecule(s) must be hitting the wall at almost the same time so that the net change in momentum of the gas is zero. This means "ideally" enough, the container would have a constant momentum, and if it were at rest in the beginning, you already know what would happen to it. It will stay in rest. Note the word "almost"! And since there is a very large population of molecules in the container, there will be many such "pairs" of molecules hitting the wall in almost every direction.

In the preceding statement, sometimes we might fall short of a few molecules hitting the wall and at the other times, there might be more than one molecule hitting the wall in a unit area ( if one molecule was supposed to hit one unit area of the wall ). So it's better to average the values over an infinitely large set of observations to obtain a near accurate result.

Time taken while calculating the pressure

Consider a single molecule. I'll consider $L$ to be the mean free path i.e, average distance between the walls or the average distance covered by a molecule in between two collisions with walls. Say, it just met an elastic collision from a wall and is moving towards another wall $L$ distance away with a velocity $v_x$ towards the wall. Change in momentum of the molecule (of mass m) when it hits the wall is $$Delta p = 2mv_x$$ (elastic collision). Now, for a time $$Delta tau=frac{l}{v_x}$$ no collision will take place and hence no momentum change. This $Deltatau$ is called relaxation time between two successive collisions of the molecule with the wall. Then after a time period $Deltatau$ it will collide with another wall and have a momentum change of $$Delta p= 2mv_x$$

This cycle will keep on repeating and repeating and repeating. So after a sufficiently long time has passed, can we say that net momentum change after a time period of $Delta tau$ is $Delta p$? So, an average of $$F= frac{Delta p}{Deltatau}$$ force is acting on the molecule due to wall. Wouldn't this be the same force that is acting on the wall due to molecule (Newton's third law of motion). Thus $$F= frac{mv_x^2}{l}$$ is acting on the wall due to each molecule.

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PS: yes, you're right about the instantaneous force. But when calculating at the bigger picture, what will you do about all the major time a molecule spends being apart from the wall? That's why we deal with average values.