Ships relying on medium energy mounts as largest and most common mount type (i.e. at least 50% of the ship's largest mounts are medium energy) versus speed and mount position:
Tempest: 180, small enough essentially both at front
Wolf: 150 + phase skimmer, nose mounted
Hyperion: 120 + phase teleporter, nose mounted
Medusa: 100 + phase skimmer, middle of ship - although relatively round shape instead of long
Shrike: 100 + plasma burn, front/broadside ship
Brawler: 100 + plasma jets (225 peak), nose mounted
Fury: 90 + plasma burn, front/broadside of ship
Harbinger: 80 + phase triple speed, nose mounted
Aurora: 80 + plasma jets (205 peak), nose/front of ship
Falcon: 80 + maneuvering jets (130 peak), middle of ship
Doom: 75 + phase triple speed, front of ship
Eagle: 50 + maneuvering jets (100 peak), middle of ship
There's is a number of distinct trends here. One, really high maximum speed for that class of ship, when maneuverability systems are included. The place their medium energy mounts very close to the front, if not on the nose of ship, which helps with range.
The Eagle bucks both of those trends. The Eagle has two-thirds of the speed of the next slowest ship on that list (which is the Doom of all things). It does have a maneuverability system, but it adds the least amount of burst speed out of all maneuverability systems. Assuming it's used on cooldown, an Eagle's average speed is 75, and peak is 100. It can't even catch some destroyers with the system on, let alone frigates, so it can't really bully smaller ships or punch down effectively. It's mount positions make short ranged energy weapons effectively even shorter ranged, at least compared to other ships.
So what is supposed to go into medium energy slots? Either beams (long range, soft flux, low dps, efficient flux use), or blaster types (short range, high damage, high flux use). Or I suppose PD, but heavy burst lasers aren't worth it at 11 OP, at least when compared to the efficiency of flak at 8 OP, or even downgrading to 7 OP for the burst laser.
The reason being ships which are primarily reliant on medium energy weapons are fast, and typically have good flux stats. So, long range hard flux is out, otherwise other ships would be easily kited to death. Similarly, even with soft flux, they can't do too much shield damage, or run into the same situation. So for the most damaging beam against shields, you have the graviton beam, which deals a grand total of 100 kinetic dps. Which, I'll point out, is the dps of a light autocannon and about the same flux/second.
Which is a problem in a slow ship, since beams are intended to be support weapons mounted on hard to catch fast ships, that very slowly build up flux on the target ship. If you're a line ship dealing half the damage (in soft flux no less) to the opponent, they're just going to run you over.
As for the blaster type weapons, the mounting point on the Eagle reduces their already short range even further. Against faster ships, the use of such short range weapons is dictated by the other ship.
The original idea behind the Eagle is clearly to mix beams and ballistics. The default designs in the data/variants/eagle all have ballistics with either graviton, ion, or phase lances. The problem is, beams aren't that great, unless heavily massed. Just one or two slow ships with 50% of their weapons in beams is simply not very good. Although ion beam is probably the best option of those available to mix with the ballistics, although due to flux inefficiency and lack of actual killing power, is best limited to 1.
Fundamentally, the Eagle is a 22 DP cruiser that in most situations has roughly the firepower of 10 DP destroyer (i.e. Hammerhead), although with the survivability and speed of a slow cruiser. It's more of a support ship when using beams, but it's relatively expensive DP wise to use in that role. It is potentially long range, but low DPS, and thus needs to be massed to be effective. It has no way to secure kills, since it has a slow and steady damage profile. I suppose if it can get an ion beam arc to knock out engines, but typically by that point, the enemy ship i already backpeddling with it's shield flux on the high end, so in line situations it won't necessarily be able to follow up.
So it's a slow, relatively tanky, support-ish line ship, although I'd typically prefer an Apogee in that role. Plasma cannon is front mounted, and does roughly the DPS of 3 medium ballistics with similar range profile (or more DPS than 2x Hypervelocity + Heavy Mauler). And then has a large missile (Squall, Hurricane, or Locust) which in most situations is going to be more helpful than 3 medium beam weapons. Champion has a better weapons profile if you really want to mix energy and ballistics, and can secure kills between High Energy Focus and missile burst.
As for fighter bays, it's easy enough to mod one in and see the effects. I'd be interested to see some fleet effectiveness comparisons. I worry a single bomber wing is typically going to be flying in by itself, and thus extremely vulnerable to PD. If I've got a cruiser with only 1 fighter bay, I'll typically be filling it with mining pods or Xyphos. It would certainly improve it's simulator performance, but I do wonder at the typical fleet performance. I suppose the initial attack wave would be simultaneous from all fighter in the fleet at least.
I'm not really sure what the right way to tweak the Eagle is, if we want to keep it's identity as the medium beam and medium ballistics line cruiser, since that identity isn't a very good one, except perhaps against fighters, which beams are good against. The Champion does a better mixture of energy and ballistics with it's large energy mount plus high energy focus. I tend to be in agreement that higher base speed would probably help. Right now, if you compare it to a Champion (speed 60), Grypon (speed 60), or Heron (speed 80), it's like the eagle has 1/2 a ship system, with half the benefit just bringing it even with typical mid-line speed profiles. Hilariously, it can't even catch the Heron in terms of average speed, even though the Eagle has a mobility system, and the Heron effectively has longer range with its fighters/bombers. And as noted by others, the Conquest is base speed 45.