I understand that pressure increases with depth in a liquid and that it is the difference in pressures at the top and bottom of an object submerged in a liquid that provides the buoyant force. What I do not understand is how the liquid exerts a pressure on the bottom of a completely submerged object. Isn't all the liquid displaced at this point? (No liquid remains underneath the object???) And if the pressures exerted horizontally are equal and opposite, it seems there is a net pressure directed downward. How then is the buoyant force still dependent on pressure differences and directed upward?
An object being completely submerged does not mean that there is no liquid underneath it - it means that no part of the object is poking out above the liquid.
If you push the object all the way down so it is in contact with the bottom of the container, you still might have some liquid underneath it because rigid objects tend to have variations in shape and surface texture which may make it impossible to remove all of the liquid below.
If your object is sufficiently soft and pliable to push out all of the liquid underneath it, then you are correct - there will be no liquid pressure pushing the object upward, which means there will be a net force holding the object to the bottom of the container. What you are describing is a suction cup, which works both in liquids and in the air by exactly the same mechanism.