Also what about Newton's third law? Wouldn't buoyancy be flawed because a balloon also displaces the air that is on top of it thus a downward force?
The pressure caused by air is a function of altitude.
Think about it: If there were less pressure below the balloon, then the air around it would rush in and fill the lower pressure until it's equalized.
In other words, the air is not an isolated column above and below the balloon. There is also the air adjacent to this column which exerts pressure; effectively, there is "more air" below the balloon than above it, in terms of air pressure.
Since there is more air weighing down on the lower altitude, the pressure below the balloon is greater than the pressure above the balloon.
Speaking of buoyancy, it's the same principle to explain why objects float in water. The greater pressure below the balloon is the result of a force exerted in all directions—including upwards. The pressure pushing downward, opposing the upward force, is the weight of the air above the balloon plus the weight of the balloon itself. Helium is about 15% the density of air (0.179 vs 1.225 g/L). That volume of air displaced by the balloon is contributing less force than normal air would. Therefore, the downward force is less than the upward force, so the balloon experiences a net force upward and floats up into the sky.
If we had an object denser than air, say a block of iron, then the lower pressure above the object, plus the density of the iron, would exceed the air pressure below, and so it would fall.
You're not looking at just the column of air that the balloon is in. You also need to look at an adjacent column of air.
Now compare the weight of the two columns of air at the same height as the bottom of the balloon.
Which is greater? Obviously, the one without a balloon shaped hole in it.
If this were a regular balloon, that hole would be filled with air and the weights would be essentially the same, but it's helium so the weight due to the gas inside the balloon is a lot smaller.
So the adjacent column of air is pushing up harder that the column of air above the balloon. It therefore rises.