Why does the arch of aorta coils?
The main question can be answered in a very dumb way: because the lower part of the body also needs blood... and this configuration is the surest way of doing that because of reasons given below (among many others, I am sure).


If you want an anatomical reason, well the most pertinent one would be that embryologically, the heart and vascular system are derived from a single straight tube, that then begins contracting and folding over itself to form the heart chambers, the aorta, and the pulmonary trunk.
If you want a physiological explanation, the aorta curves slowly because it is a point of the circulatory system where pressure is high. Apart from the fact that sharp kinks would cause a massive and useless loss of kinetic energy on blood flow from shear stress over the aortic walls (and therefore more heart workload), the induced shear stress would also damage the aortic walls much more quickly than in the actual configuration.


The third reason is just theory on my part, but it seems the vascular system is designed by priorities: you could imagine a system with two aortas. One for the upper body going up, and a second going down. However, such a configuration makes it mechanically impossible to balance perfusion to the organs, and especially ensure brain perfusion. For example, how would the body compensate if the upper aorta got clogged while the other stayed wide open? The single aorta configuration allows the heart to adapt to such mechanical perturbation scenarii using, for example, left ventricular hypertrophy.

However to add to some elements of your question:


the reason of the aortic coiling has certainly very little to do with elasticity and the vessel compression chamber principle.
Higher elasticity means the aorta is less stiff, not more.
When the aorta gets stiff (elasticity decreases) as it is often the case with age and calcification of the aortic walls, it is correct that the heart will need to work more, that systolic pressure will get higher, and diastolic pressure lower. This gives the classical clinical symptom of "cannonball" or "gunshot" pulse