Hiruma Kai's point of the Eagle effectively not having a ship system is bang on, as are all the other poster's comments about hard flux medium energy weapons just not fitting the Eagle's size/role. Hard flux medium energy weapons are for close range hunter ships, and with its new faster speed the Falcon can use them effectively but the Eagle has trouble.
Advanced Optics + Phase lance is a good combination and even gives the ship some burst damage, but has a few problems:
1) OP expensive for the hullmod, and while the hullmod also helps beam PD get more range, it slows down their turn rate tracking, so advanced turret gyros might also be needed for even more OP.
2) Even with AO the energy turrets are just set quite far back so it loses more range (this isn't a huge problem, just annoying)
2) AI has a burst fire bug where if it turns the autofire group off mid burst the guns actively turn off target (fixed hooray!)
3) AI has a nasty tendency of wasting the burst by firing just outside of range/into empty space vs non-circular ships - its aiming for the enemy's collision radius rather than hull.
4) If the AI activates its system and decides to spin during a burst the turrets can't keep up and the burst is wasted. This also applies to the front kinetics.
Even with these problems kinetics + phase lance is an "ok" Eagle build. Its decent, just not exceptional in any way other than occasionally flash popping swarming frigates.
The other successful usage I've had of the medium energies on Eagles is beam support. Ion beam behind kinetics is quite good for supporting other ships, but it doesn't kill things, and the Eagle doesn't do the job much better than a Falcon would. Gravitons have decent efficiency but their high OP cost/damage and lack of hard flux make them very low impact expensive weapons, to the point where the slot is less than half the value of a medium kinetic slot. A major buff to beams would help make Eagle's more useful - we'd need to be careful about making the high tech ships too strong with them, but currently beams really aren't a good build on them other than the Disco Paragon so I think there is room to do this.
But going back to the system: Maneuvering Jets + forward facing kinetics are a double edged sword. On the one hand, it lets the moderately sluggish ship keep its guns on target vs smaller ships. On the other, I think that while its on it makes the AI 'think' that its more maneuverable than it is: it then tries to use that maneuverability to either distribute armor damage or swing to a new target, then the system turns off, and the Eagle is now "stuck" for a little while not being able to shoot its main guns at anything. The same thing happens to the Falcon a little, but again the Falcon's higher base maneuverability means that it isn't stuck in the bad position for nearly as long.
Giving the Eagle better base speed and maneuverability and then an offensive ship system would go a long way to making it competitive. While new systems would be cool, what if there was one version that got AAF, and one version that got HEF? Say for a certain purple painted faction that has special energy weapons?