How do pulleys redirect force? Consider this setup of three blocks, in which you would like to find the acceleration of the 8 kg one:

One way I've heard to solve it is to consider the three blocks as one system and look at the forces exerted in the direction of motion and against it, which would be the 40 N force of gravity and the 28 N force of gravity respectively. However, if I were considering the three blocks as one system, I would have thought to add the forces acting in the same direction as the block, which in this case would mean adding the two forces of gravity, not subtracting them. To me, considering the forces relative to the direction of motion seems to indicate that you simplify the system to be a linear one, so that the two forces of gravity act in different directions:

Mathematically, this does seem to be a valid operation, as in either case, the acceleration of the 2.8 kg block is $$a= (T_1 - 28\text{ N})/2.8$$, the acceleration of the 8 kg block is $$a = (T_2 - T_1)/8$$ and the acceleration of the 4 kg block is $$a=(40\text{ N} - T_2)/4$$ Since all three accelerations are equal you have 3 equations for a 3 variable system of equations and one solution for acceleration, therefore making the acceleration the same in both cases.
Conceptually is where I have trouble with this idea. I assume it is the pulley that allows you to make this "linearization" of the system because I've heard a pulley "redirects" force but why? What forces does the pulley exert on the three block + two rope system allowing you to treat the system this way? And if you did consider the forces exerted by the pulley, would there be a solution to this problem in which you add the forces based on their direction, not based on whether they act with or against the motion of the system? In other words, something like this:

 A: Yes, if the tension in one of the strings is $T$ and the string is vertical on one side of the pulley and horizontal on the other side of the pulley then the pulley must exert a force $\sqrt 2 T$ at an angle of $45^o$ to the vertical on the string. This force is how the pulley "redirects" the tension in the string. And by Newton's Third Law, the string must exert an equal and opposite force on the pulley.
Similarly, if the string goes vertically up on one side of  the pulley and vertically down the other side, then the force exerted by the pulley on the string is $2T$ vertically upwards, and the string exerts a force $2T$ vertically downwards on the pulley.
However, in string and pulley problems we are not usually interested in the forces exerted on or by the pulleys, so we just assume that each pulley exerts whatever force is needed to "redirect" the tension in the string.
A: 
Conceptually is where I have trouble with this idea. I assume it is
the pulley that allows you to make this "linearization" of the system
because I've heard a pulley "redirects" force but why?

An example where a pulley only serves to "redirect" force without reducing the force due to a mechanical advantage, is a the following.
Supposed you need to lift a weight. To lift the weight you need to exert and upward force, i.e., you need to pull the weight up. As an alternative , you can use a single fixed pulley with a string over the top with the weight at one end of the string and you applying a downward force on the other end to lift the weight. For a massless frictionless pulley and string, the pulley has served to "redirect" the force needed to lift the weight. So why would one do that? By redirecting the force you can use your own weight to help pull up the weight. That's easier than only pulling up with the use of your arms.
Another example is in the one you have given. The 8 kg weight could be accelerated by applying a net horizontal force without the use of pulleys or a string. Since the ideal string tension on each side of the 8 kg mass is constant throughout, the pulley changes the direction of the tension force from vertical to horizontal.
Hope this helps.
A: The "considering it as a linear system" arises because, by design, no component of the system can move other than in a linear manner, in effect (unless one of the blocks turns a corner at the edge of the table).
The mass at the tabletop can only move left to right.  The masses left and right can only move up and down, but because of the (inelastic) string, a movement up and down correlates with an identical movement of the top element left and right, and the other sides mass moving down and up.
Similarly, the tensions in the string pulling up, is the same as the tension left and right.
How can this be? Because unless something breaks, the table and pulleys will exert whatever force is needed, in any other direction, so no other net forces exists.
