Because downward force is not isolated from the upward buoyant force.
if the object is lighter than the fluid it displaces it will be pulled higher. Just the same, if the object is heavier than the fluid it displaces it will be pushed lower.
If the object doesn't change its volume, it's displacing the same amount of air regardless of pressure.
So, say location A has low pressure atmosphere and an object displaces X cubic whatevers of rarer, lower pressure air.
Location B has high pressure atmosphere and the object there displaces the same X cubic whatevers of denser air.
Which locations' displaced volume of air weighs more?
Location B, right?
At which location does the object with X volume experience a greater buoyant force?
Same answer. Location B. Right?
Yes, higher pressure air "pushes down" more from above, but the buoyant force also "pushes up" more.
This is a key principle in scuba diving. Buoyancy compensators adjust the diver's volume to manage the amount of water being displaced as water density/pressure changes with depth. Great pressure with greater depth squeezes the diver and his neoprene, making him less voluminous and less buoyant. But adding air to the BCD increases volume and increases buoyancy by displacing more of the higher pressure water. If an object is non-compressible, its buoyancy will increase with increase in water pressure. The same is true of the atmosphere. High pressure air increases buoyancy, as long as volume remains constant.
The error is believing that fluid pressure is just a vector force in one direction. Bouyancy isn't "something else." It's part of the pressure force equation.