As I have mentioned before, we had a saying in the mid-eighties where choosing a CDS is concerned:
"If you want good leadership, pick a general; good administration, pick an admiral; good politics, pick an Air Force general."
Retired general Hillier almost choked on his coffee when I told him that one a few years ago and told me: "You have no idea how true that is". I believe that this is a reflection of the element we come from.
There is no doubt in my mind that by its very nature, the army develops in their general officers leadership skills that are at the summum of human leadership. Commanding large formations of soldiers motivated to fight while only glimpsing their "general" from afar occasionally and hearing only a very few of his actual words, if ever, requires no less.
Navies, on the other hand, are extremely complex industrial undertakings. Anyone walking about the Dockyard can see that. Navies were dealing with engineering, naval architecture and worldwide communications/command and control of forces issues when armies were still marching by foot or horses, on their stomach, and carrying their "command" element to make all the decisions locally. Lord Nelson was the hero of Trafalgar, but everyday, he sat at his sea desk with his secretary to work through piles (already) of "correspondence", which even then could have included anything from authorizing a captain to exceed his allowance for local purchase, to suggesting a modification to the design of some part of the ships, to reporting to the Lord admirals on his next plans for pursuit of the enemy. As a result, navies know the importance of good administration (and in my experience, on average, Navy units always came out better than Air Force and Army ones when administrative reviews or inspections are carried out ).
As for the Air Forces, regardless of the fact that they put together squadrons, wings and groups, it remains that in effect they fight either alone or in small elements, with a distinct divide between the flyers, in their officers world, and the ground support crew living in the non-commissioned world. Thus, they are part of a fragmented type of organization, hence their love of functional command (fighters, transport, patrol, etc. each flying in the same area under different commanders) which is abhorrent to Army and Navy who prefer geographic command. Since little distinguishes one pilot from another, failing a war where you make Ace, their career progression must rely on their interpersonal skills, hence the good "politics", which is used here in its wider human sense than political parties.
This said, I think it accounts for the lack of admirals as CDS. Good administration doesn't get you noticed. It is one of those things that, done right is invisible, and is only remarkable when you screw it up in a big way. Add to that the fact that the Navy is the "silent" service, skill it developed as it knows that it can lose wars, but it cannot win them*, and you have all the ingredients to be passed over regularly even if you have all the skills for CDS.
*: Nelson won at Trafalgar, but it was Wellington who beat Napoleon at Waterloo that ended the war. The longest battle of WWII was the Battle of the Atlantic, from day one to day last, but it was the massed armies of the allies that defeated Hitler. The RN quarantined the Falklands and cut off Argentinian supplies, but it was the soldiers and Marines of the U.K. that reconquered them. In each of those cases (and countless more) the Navy won its battle but it didn't defeat the enemy -the Army did. Had the Navy lost those battles, however, the enemy would have won in all cases.
