I think you've missed the point. Pardon if i was rushed and the explanation is brief.
The sector admiral is in the thick of operations at the depot not at the sector capital. However, I do believe the Imperium Admiralty is located at 3I "Capital" and not Core depot (where the sector admiral would be). It is not a cushy job. There are noble liasons to deal with nobles and governments.
Each sector depot has its own security fleet and military assets. I would assume (it is not documented) that an sector admiral can move any assets he needs. If this admiral is going to leave the depot I believe they can utilize whatever asset is appropriate. During peacetime this SA may use a capital ship and during war, they may use a squadron or lead the sector fleet, as is documented in 5th frontier war history, against an adversaries.
Also note your example may not fit well. Eisenhower used an airplane not an entire fleet or squadron. He had a fighter escort.
Let's clear up the "depot" bit first. The nearest depot serving the Spinward Marches is in Deneb sector, very far from the thick of operations. 21 parsecs from Rhylanor, in fact. I think canon mentions someplace the Marches keeps retired ships for activation as replacements in wartime, but I don't know of a depot per se in the Marches. Lots of naval bases, no naval depots that I know of. For the Marches, the most likely center of naval operations would be Rhylanor, far enough back to not be overwhelmed in the opening moves but close enough to provide coordination and administrative oversight.
Thus, while the sector admiral of Deneb might indeed be located at Depot/Deneb, the sector admiral of the Marches has no depot.
As for what a sector admiral does in war, I can't speak authoritatively for canon beyond CT and MT. I have no Mongoose - an omission I'm thinking about correcting - and I have only a few GURPS books, none being the right one for this issue.
Spinward Marches Campaign and the various
Journal news items speak of Santonocheev, and later Norris, in an organizational capacity, deciding grand strategy and issuing orders to entire fleets. I don't see anything there that speaks to their actual presence at any given battle, though there are a lot of
Journal news items and I may have missed one. That is of course absence of evidence, not evidence of absence.
However, absent evidence, I fall back on MT's
Rebellion Soucebook, which tells me, "At their lowest level, Imperial ships are organized into squadrons (from three to 10 similar ships). Squadrons are grouped into permanent numbered fleets (usually three to 10 squadrons per fleet), although squadrons or individual ships may also be detached into temporary task forces for specific missions. Numbered fleets (usually within a single Imperial sector) are grouped into named fleets." It goes on to say, "A numbered fleet is a group of several squadrons; a named fleet is an assembly of two or more numbered fleets."
This tells me operationally that a numbered fleet would have an admiral responsible for that fleet and the sector admiral would in turn have all those admirals answering to him. The sector admiral would not have a fleet of his own unless he was also admiral of one of the numbered fleets. While it can be dramatic to have the head of the sector in charge of one of the spears, operationally it's much like having the CEO of Wal-Mart being also responsible for the operation of one of their stores, or maybe one of their city branches. It draws that leader's attention away from his core responsibility, which is to coordinate all those subordinate executives, or admirals in this case, so they're all acting to the same overall game plan, and with communication lags of weeks or months, effective coordination is vital.
I don't know what Mongoose says, but if Mongoose says different, it's badly flawed canon. It is very poor organization to distract the head of a major organization by also making him responsible for one of the subsidiary branches. I do know that I can't find anything in CT that definitively puts a sector admiral at the head of a numbered fleet, not outside of civil war references, and that's - well, if you're going to cross the Rubicon, you're going to have to do things differently anyway or you'll be undercut by the loyalists in your own ranks.
There's enough wiggle room for some gamemaster to do things as he chooses, but this business of giving the sector admiral his own fleet - again, it just distracts him from the job he has. Even if he were co-located with a fleet, as with the depots, I'd put that fleet in the hands of a fleet admiral and make the fleet admiral responsible for managing it. He's got to obey the sector admiral's orders anyway. Only way I'd give the sector admiral his own fleet is if I suspected the potential for insurrection in the ranks and wanted the sector admiral to be able to battle one of his own subordinate admirals. And that's pretty much what it says: the sector admiral doesn't trust you, so he's taking vital war equipment out of the war effort and reserving it for his own use just in case he needs to use it on you. Up until Dulinor, there hadn't been any indication of that potential within the Navy for centuries.
I've considered a number of alternate examples, but they come from post-radio and don't really answer the issue of administration in the presence of significant communication lags. However, given that we're dealing with coordination of of a dozen or more fleets totalling a thousand ships scattered through a few hundred star systems, I would consider the duties of the office to be too demanding to let the sector admiral be distracted with the operations of a single numbered fleet.