The Arachidonic acid “cascade” is a complicated series of balanced and dependent reactions that begin with a substrate and produce products.
Arachidonic Acid is the substrate required for the reactions. I am unaware of any circumstance where there has been a deficiency in the basic production of arachidonic acid as it is so readily made from other molecules. It would seem to me (I do have a Ph.D. FWIW) that supplementing specifically arachidonic acid is akin to supplementing glucose. It is just something your body will maintain production of (even at the expense of other metabolic functions).
The two main pathways of Arachidonic acid metabolism are (as you pointed out) the cyclooxygenase path and also the lipoxygenase path. Most common drugs that interact with this system do so on the cyclooxygenase path (ASA, NSAIDS, Cox-2 inhibitors). There are some drugs -usually for asthma- that work on the lipoxygenase pathway, such as the leukotriene synthesis inhibitors.
At this point, if you have a pathology there is some utility for using these drugs and certainly the prophylactic use of low dose aspirin for prevention of atherosclerosis has been shown. However, like messing with the androgenic cascade with exogenous steroids, pushing/inhibiting one arm of the metabolic process causes an increase in the other (ex. increased estrogen with exogenous steroids). In this case, inhibiting cyclooxygenase often leads to an increase in the lipoxygenase product and vice versa. This may be a cause of the Viox debacle but that is a novel unto itself.
All in all. I think Arachidonic acid is far too ubiquitous and the the ability to finely control the enzymatic process far too limited to worry about at this point for general health purposes.