is it likely that a navy pilot (iceman) could rise to the rank of Admiral?
Sure - if he had squadron command as an O-5 Commander, was successful in that, was selected for promotion to O-6 Captain, and was then selected for major operational command as an O-6, for something like air wing command (commander of the air group, all the squadrons which deploy on a carrier) or aircraft carrier command, which means he would have had to go to nuke school along the way. If he was successful in that, he’s in play for flag rank.
The pyramid narrows abruptly and drastically at the top. So many superb captains with outstanding performance in command, and the Navy will choose the ones with the skill sets it needs in the next crop of future admirals.
There are admirals in every officer designator community, sometimes not very many, as the number is proportional to community size and admiral roles in the community. For example, the JAG community is relatively small, they are a supporting staff role, and I think at any one time, they might only have 1-2 admirals (that was always my impression). The heavy-hitter 4-star admirals come from the big unrestricted line communities (surface, air, subs, etc.) to command the major operational and fleet commands, joint commands, NATO commands. The 1, 2 and 3 star admirals are sprinkled about.
Finally, many do not realize an unrestricted line officer has their operational specialty which they hone at sea, but also a sub-specialty, often related to their Master’s, in which they develop expertise ashore, for all the skill sets needed to run big Navy. Sub-specialties might be financial management (worked in budget and programming roles ashore), manpower management, training, strategic planning, operations management, requirements analysis, cyber operations, international affairs, Congressional liaison, etc. That’s often critical as the Navy looks ahead and determines what flag officers it needs in its future group of admirals.
As an example, you can see that VADM Stockdale had command of a fighter squadron and an air wing.
en.wikipedia.org