Chambering is not the correct way to kick. Chambering is a method to teach the correct balance and general way kicking is to be done. Look at any true masters of the martial arts. You will not see them chamber any kick or strike/punch. The problem is that people are taught to chamber, but never taught that after they learn proper technique and balance they should not chamber any longer. Chambering only makes any technique slower and also telegraphs the technique giving the opponent warning as to what is about to come. If you were playing poker, would you tell the opponent what your cards are and what you are about to do? I certainly hope not.

Bottom line here is either you are kicking correctly or you are not. Since I can;t correct your kicking without working one on one with you, I can offer only this.... do your kicks slowly. If you feel any binding in your hips, then you are not doing the kick correctly. Then you need to hold on the the wall so that you can do the kick very slowly and adjust the kick until you find the motion that no longer binds your hips.

...

I chamber side kicks horizontally. Piercing kicks start with the sole against the thigh. Thrusts start with the sole facing the floor.

I only know one side kick and that is the taekwondo sidekick that Bruce Lee used, keep on practicing that

Did you try to do a search on Google?

Whar are the advantages to the traditional vertical chamerred side kick?

Modern sanda fighters tend to chamber it diagonally/ horizontally which is more the direection the kick is gonna go.