You keep making leaps of logic that are not there. The UEF trusts information from Carey. That does not require them to know of her whereabouts or activites at any given moment. They don't necessarily need to know she's still a piece on the board. My argument contradicts nothing.
We still don't know if Carey was arrested before or after she leaked the Carthage's positions. Here, I'll walk through it point by point.
1a) We are
absolutely certain that Carey was the source of the Agincourt intelligence leak.
1b) We are
absolutely certain that the intelligence leaks for both the Agincourt and Carthage were the same source. (thanks for that quote, Drogoth, it had slipped my mind)
2) This means that Carey must have been both leaks; there is not another posibility, because the briefing uses a singular third person pronoun; there is no they.
3a) We are
absolutely certain that Steele formulated a plan to remove the Wargods from meaningful consideration in Sol.
3b) We are
absolutely certain that the Carthage attack was part of Steele's plan.
3c) The capture of the Agincourt instilled in the Wargods a sense of confidence and invincibility, which was vital to Steele's plan to get them to attack a significant target (of his choosing).
4) This means that both information links worked toward the proper execution of Steele's plan.
4a) The reasonable assumption is that both leaks were deliberate.
5a) Carey committed treason against the GTVA by leaking intelligence to the UEF (points 1a, 1b and 2).
5b) The punishment for treason before a trial in the GTVA is to be stripped of rank and citizenship in the GTVA, and await summary court martial and/or board of inquiry(if I remember my terms right).
5c) The punishment for a guilty verdict on the charge of treason against the GTVA is highly likely to be execution. If not that, rotting in a correctional facility for the rest of one's natural life.
6) Carey is no longer a player in the Sol theater.
7a) Because of 2, 4, and 4a, it is likely that Carey either deliberately acted against the UEF, or was manipulated into acting against the UEF by some other entity, most likely Steele.
7b) Because of 1b), in which both intelligence leaks were accurate (outcomes aside), the UEF trusts intelligence from this source, which, because of point 2, must be Carey.
7c) Carey's arrest indicates that, whatever the other outcomes, the intelligence leaks were legitimate and led to damage to the GTVA.
8) The UEF will likely listen to any further intelligence from Carey.
9a) Because of 6, Carey has been removed from her post and any post that could influence the Sol theater.
9b) Steele knows this.
9c) The UEF knows this.
9d) Steele, just as any GTVA service member, knows the punishment for treason.
9e) Steele has been known to conduct under-the-table operations to secure an outcome favorable to him.
9f) Rescuing Carey from execution and/or a correctional facility puts Carey in Steele's debts.
9g) Carey is a person of strong morals and integrity (point 5a).
9h) Steele does not ignore assets that could give him an advantage.
9i) Carey is either predictable or under orders to betray the GTVA (points 4 and 4a)
10) Saving Carey, either legally or illegally, gives Steele another piece on the board, and one he already knows how to use, and further one that is in his debt, personally.
11a) Steele strives to use all tools in his arsenal to their greatest effectiveness.
11b) Per point 10, Carey is a tool in Steele's arsenal.
11c) Per point 8, the UEF will likely listen to any further intelligence from Carey.
12) Steele can use Carey to feed the UEF information that will improve his effectiveness.
QED.