If liquid pressure was the same at all depths, would there be a buoyant force on an object submerged in the liquid?
The simple answer is no because, with no pressure gradient, the upwards and downwards forces the submerged object experiences, which are due to the pressure gradient, would disappear.
More generally, the outcome for a system comprising the fluid, and the object that will float or sink in the fluid, will be determined by their respective densities. An object more dense that the fluid will displace the fluid and experience a bouyant force (equal to the weight of fluid displaced).
Behind all of this is gravity . What is happening is that the system is minimising it's potential energy, exactly in the same way that an apple does when it falls from a tree.
If you could switch gravity off, you would not observe these effects. At the same time the pressure gradient in the fluid,