How does bath application of KCl depolarize a neuron?
Membrane potential (measured electrophysiologically, or calculated using the Goldman equation) depends on two things: concentrations and permeabilities. In the Goldman equation you see this directly: concentrations multiplied by permeability, in both the numerator and denominator.

If the membrane is not permeable to an ion (PIon=0), that ion will not affect the reversal potential. In a typical resting neuron, in the absence of any synaptic activity, Chloride conductance is near zero compared to ions like potassium.

I'm not as clear what you are asking about patch clamp experiments, I will try to just describe the process and hopefully that helps.

In a whole-cell patch clamp, the inside of the cell is continuous with the patch pipette solution, which usually contains high concentrations of potassium, like the normal intracellular environment. This is done on purpose, to try to record from cells in the most natural condition that can be replicated simply.

During the process of making a whole-cell recording, however, the experimenter has to first place a patch pipette down near the cell. While doing this, some solution is leaking out of the pipette and increasing extracellular potassium concentrations, which can depolarize surrounding cells. So, patch quickly my friends, you are making it uncomfortable for everybody before you make a good seal.

(the process would be similar for inside-in patches; for on-cell or inside-out patches, one would use different ionic concentrations in the pipette to mimic the extracellular space, or these would vary depending on the exact experiment performed)

edit: Just wanted to note that a statement in your question is mostly wrong; "the pipette is loaded with KCl and injected into neurons that leads to depolarization" - KCl inside a cell would not depolarize that cell, unless the concentration of K+ was lower than the typical K+ concentration of the cell, in which case you would probably have other issues and a dead cell from osmolarity problems.