Why isn't it that we are pushed more one direction left or right than the other if we aren't standing in the middle of the pool/ any fluid? By pressure=mass(height)(gravitational constant) and pressure being in multiple directions (not only downward but also sideways) it seems like there is more pressure on your body in one direction than the other? But IRL you aren't pushed one way more than the other. Also inside a pool it's pretty static even if you are standing in it?
I find the best way to think about pressure is to imagine dividing the pool into small cubes of water and think of all the forces on a single cube.
First consider cubes in a horizontal row and the horizontal forces on each cube. Each cube has a pressure force from the cube on its left and an opposite pressure force from the cube on its right. So the cube does not move.
Now come to you. You are really just another cube. So the only horizontal pressure forces on you are from the cube to your left and the cube to your right. These are the same so you don't feel a net force where ever you stand in the pool.
Finally think about the vertical forces on a cube. Now we have an extra force of gravity acting downwards. This means for the cube to stay stationary you must have:
$$ Pressure_{up} =Pressure_{down} + Gravity_{down} $$
This explains why the pressure increase with depth.
If you were forced to the sides of the pool when standing in the middle, that means fluid pressure in the center of the pool is larger than at the sides. If this were the case, we actually wouldn't have a static fluid. The fluid itself would also be forced to the sides, just like you. This would happen if you suddenly removed the walls of the pool, and fluid started flowing outwards. You would feel the flow pressure $\rho V^2$, and flow outward with the fluid. In a static pool, however, nothing is pushing you to the sides. Why is this?
The answer is that hydrostatic pressure only varies in the vertical direction. There is no change in pressure as you move from side to side.
Furthermore, every piece of fluid at a given height has the same pressure in all directions, like you've mentioned. This is Pascal's Law. A piece of fluid (fluid particle, fluid element, etc.) near the wall has the same pressure as a piece of fluid in the center of the pool. You therefore feel the exact same pressure whether you stand in the middle of the pool or near the sides.
