The "Eligibility" trait simply shows that an FM is qualified to hold a particular office. There is no requirement to send them to Carthage unless you want to. As for faction leader eligibility for a particular office, keep in mind that there was no single leader for this faction, but the M2TW engine forces us to designate one - so even though it might seem odd for an FL to perform a lower level office, in historical reality that guy would have been just one of many leading citizens.
That said, there is a lot of work planned for Carthage in v2.4 (the release after the pending v2.35), and I'm sure we'll take another look at the way eligibility is handled and offices are awarded.
For a more detailed description of the various offices and election procedures,
see this post.