Yes and no. I agree it is a very niche unit due to its placement and how Aztec plays in general.
I tend to find that if there is any possible nearby civ to bully early you should do it. But this means that pretty quickly you'll likely have taken workers from civs/CSs, gotten plenty of gold to buy others, etc., and Aztec plays pretty tall in general too so you don't need 10 workers for your 5 cities. Getting workers is cool, Sentry is fine, not needing Iron is solid for Aztec's start, but it only has +1 CS which is a ~7% buff, and doesn't have any of the Jaguar's very powerful abilities. So overall it isn't any better at bullying civs than a swordsman would be, it just rewards you a bit more for it, and it doesn't need Iron which is admittedly great.
Essentially, Eagle is an okay unit in a really poor spot on the tech tree for Aztecs that doesn't give enough to be worth it imo, with a few exceptions. Also it feels awkward when you've got two melee units competing for the finishing blow, and it feels taxing on your unit supply to have both UUs be melee one after the other. It is a nice way to have a Swordsman unit that doesn't require Iron and having a million workers can be fun assuming you don't get screwed by the maintenance.
I don't think they should upgrade into each other, that would be insane (and historically inaccurate, but so is a lot of how the Aztecs work in game). Any enemy near a jungle would get blitzed by ultra-promoted Eagles that would smash any unwalled city. And the flavor of the Eagle is fun enough. Maybe move the Eagle to a later unit like Longswordsman? The Aztec Empire actually lines up historically with the Renaissance, from early 1400s to early 1500s being their time of power. Or slot it into a different tech on the tree?
The issue I see is how to fit the two units properly into Spearman/Swordsman lines without overpowering Aztec's early game but still making both units powerful. Too much strength in Ancient warmongering is way too good when civs won't get walls too quickly.
Total side note, is it a good idea to update to the newest VP patch at this point? Not sure if it/4UC have gotten to a point where at least I won't get massive issues.