Midline is my favorite tech. I did a Midline playthrough in 0.95a that worked really well. Haven't progressed in 0.95.1 far enough yet to use Falcon/Eagle in my fleet (been using Eradicators like everyone else) but here's my take on them.
Eagle:
One of my favorite Flagships in midgame was an Eagle (XIV) with Heavy Autocannons and 2x Phase Lance + Advanced Optics. Strong Kinetic pressure followed up by powerful anti-armor. 800 range Phase Lances are awesome, and quite good at sniping Frigates. The flux buildup is terrible though - hence 2x Phase Lance rather than 3x. Definitely a playership-only build.
Then one day I realized Eagle (XIV) has 46 base speed, which is only 1 (!) more than Conquest. And a Conquest brings a lot more firepower to the table.
As a playership, the Conquest is just better than Eagle. In a fleet, by DP you could have 2 Eagles against 1 Conquest which looks more even. In terms of firepower though, that single Conquest has some advantages. More range with Capital-grade ITU and Gauss Cannons instead of HVDs. And much more Killing power with 2 ECCMed Hurricane MIRV and 2x Harpoons. In the support role, Eagle faces stiff competiton from Conquest.
The one advantage of Eagle over Conquest is durability. Two Eagles can tank a lot more damage than one Conquest can. Enter the Champion, a ship with similar DP cost, similar flux stats, a good bit more armor, and much more devastating firepower with a large Energy + HEF and a large Missile. As a ship-of-the-line the Eagle is completely overshadowed by the Champion.
So yeah, the Eagle has been obsolete ever since 0.95 gave us the Champion.
>>> To make the Eagle relevant again, I say give it +10 base speed to reinforce its role as a Fire Support ship that is difficult to flank.
Falcon:
The Falcon attempts to fill two roles.
> One: an "Eagle-Light" that provides fire support with HVD/Mauler/Graviton Beams against enemy Cruisers and Capitals. The advantage of the Falcon over the Eagle is it has enough speed to keep its distance from enemy Cruisers, which can be very useful.
> Two: a hunter of Destroyers and (slow) Frigates. The Falcon can overpower any Destroyer in 1v1, and has just about enough speed with Maneuvering Jets to give chase.
Enter the Eradicator and its (P) variant.
An Eradicator with AAF and 2x HVD + Mauler provides more firepower than a Falcon can muster, and also has enough speed to keep its distance from enemy Cruisers. So it provides very stiff competition for the Falcon in role one.
The Eradicator (P) with cancellable Burn Drive can chase Destroyers and Frigates just as well as a Falcon can, while also having 50 more flux dissipation, much better weapon mounts and 25 more OP to fill those mounts. So it provides very stiff competition for the Falcon in role two.
Indeed, one could say the Eradicator has obsoleted the Falcon.
>>> To make the Falcon relevant again, the changes proposed by Alex seem reasonable.