My two cents on the eagle is that they're in a hard spot along with the other mesoamerican civs because your only replacement options are infantry and recon troops, and only from ancient to medieval, really.
Strictly speaking, plenty of the native American civs are completely misrepresented time-wise in Civ 5. Though you are correct that it is difficult to have a proper set of UUs when none had significant military navies and only the two North American native civs had any mounted military units. The Aztecs descended from people who migrated in the 1200s and then the empire reigned from early 1400s till early 1500s, placing them in Medieval, definitely not Ancient. Incan is also largely misrepresented, their empire was largely concurrent with the Aztecs, although they descended from earlier civilizations as early as 300; still they should be at least Classic through Medieval, not Ancient. Iroquois should be Medieval through Renaissance.
Maya, Polynesia, and Shoshone are all well-enough represented as far as time periods go.
So really if we wanted historical accuracy we should have:
Polynesia spanning Ancient through Medieval
Maya spanning Ancient through Renaissance with a focus on Classic
Shoshone focused on Renaissance with some earlier since we don't exactly know when they started
Aztecs Medieval
Incan Classic through Medieval with a focus on Medieval
Iroquois Medieval through Renaissance
So really we have plenty of space for these civs, especially since we have Polynesia on its own, Shoshone and Iroquois in North America, Inca in South America, and Maya and Aztec in mesoamerica. The only thing that is really awkward is Aztec because base Civ 5 stole Atlatlist, so if we want Aztecs to technically be historically accurate with their 4UC UU, it should be a Medieval ironless longswordsman. That is the time period that the Aztec Empire was dominant, and represents the lack of anti-cavalry but also ironless melee infantry.
As for what they should do, creating workers is fine, you can always sell them and this is roughly the time period you'd likely go crazy with offensive wars since you just got the Huey and also have realistic options to capture cities now that you have strong siege units and an infantry to protect them. And new cities means you need workers. Though I did like the idea that the captured workers work at full strength I can take or leave it.