The pressure isn’t just pushing the bubbles down, it pushes equally on all sides. If the pressure inside the bubble is less than that of the surrounding water, then the bubble gets compressed down until the internal pressure matches the external pressure. It’s just a balancing of forces.
Buoyancy is a phenomenon that actually has very little to do with pressure, and the amount of buoyant force you experience is dependent on the mass of the fluid you displace, which is dependent on the volume you occupy and the density of the fluid you’re in.
If you want to change the buoyancy of an object by changing the pressure, you need to make sure that the object in question compresses differently than the fluid around it. Like a Cartesian diver has a little air bubble in it, so when you increase the pressure, the air bubble is compressed more than the surrounding water, causing the diver to have more density than the surrounding water, and sink until pressure returns to normal.
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