Damn man, it's like you read my mind. This is exactly what I wanted to say just hadn't had the time to say it yet xDI think there's a few issues that aren't necessarily directly related to the strength of the eagle unit that makes going eagles a strong choice. Most civs don't want to tech into longswords as a long game plan, where as most meso civs do want to tech eagles. Food eco requires more effort to set up than gold eco. Longswords are easier to counter than eagles and really only have one purpose. Eagles are a good unit for controlling the direction and pace of the game while longswords aren't. The burden is often on the non meso civ to find out what the other guy is doing and react.
I think an ok change could be to slightly reduce the bonus eagles get vs cav. No one really goes eagles to counter kts, they just happen to be efficient enough vs knights that you don't have to change tech. Scouts become more useful and kts more viable. This could be enough to make the meso player also want to add pikes or xbows or something to make it a little bit more interesting and more balanced in terms of techs and eco required while also offering the non meso guy a unit path that allows for raiding and map control.