Why isn't the pressure of the liquid uniform throughout?

There is also the pressure due to the weight of the water.

I had this argument in high school physics class (a great many moons ago).

The facts are that the pressure exerted by a contained liquid is

P = rho g h where rho is density and h is depth.

Now if additional pressure is exerted on the liquid, by the atmosphere for instance, then that ADDED pressure is spread evenly throughout.

So there is still a pressure gradient over the depth of the liquid.

We could summarize by letting external added pressure be Px then

P = rho g h + Px at any point within the liquid.

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According to Pascal's Principle, in a uniform fluid, pressure depends only on height. There can be no pressure gradient except for that caused by liquid pressure. This is seen in hydraulic jacks which are used to lift heavy objects.