The eagle is historically considered the king of birds, in the same way that the lion is the king of beasts, both being apex predators of their respective environments. It's no surprise that early kingdoms and empires latched onto them as a symbol. You noted the (first) Persian Empire, ancient myths about the himself possibly mythical forefather of the Achaemenid dynasty, Achaemenes, had him being raised by eagles, though the earliest reference I've seen to this is the 2nd-3rd century Roman author Aelianus:

I have heard however that Achaemenes the Persian; from whom the Persian aristocracy are descended; was nursed by an eagle.

In Greek myth, the eagle is connected with a fellow by the name of Zeus, being the form the God of non-consent took to carry off Ganymede to serve as his cup-bearer, according to Homer.

Fast forward to the late Roman republic, the eagle was one of many symbols used as symbols by legions (not on banners yet, this early on we're talking about having a guy carrying a small statue suspended on a staff), Pliny the Elder specifies that these included the eagle, ox, wolf, horse, and boar. After a particularly thorough loss in the Battle of Arausio in 104, the Consul Gaius Marius revamped the Roman military, most notably to this discussion cutting all of the other standards except for the eagle, making it the sole symbol of Roman military might.

It's also important to lay out the religious symbolism. Eagles appear throughout Jewish and Christian texts, but notably here we're considering the Books of Ezekiel and Revelation, in which both authors claim to see God accompanied by man-like cherubs with with faces of animals. Ezekiel identifies them as each having four faces, the face of a man, lion, ox, and eagle, while John of Patmos only claims one face on each, but including each of the named animals (although specifying a calf rather than an ox). These four animals came to be associated with the four authors of the Christian gospel as early as the 2nd century AD by the Greek bishop Irenaeus, although the most common associations were proposed by Victorinus, who proposed associating the eagle with John, author of the book of and epistles of John and possible author of the book of Revelation.

So, with the eagle having strong symbolic connections to strong empires, military might, and Christianity, it should come as no shock that it was adopted as the heraldry of later empires. Charlemagne had a bronze eagle placed on top of his palace at Aachen. By the 13th century it had morphed into the double-headed eagle more commonly associated with the Holy Roman Empire, being depicted as the arms of Frederick II by Paris' Chronica Majora.

When the German Empire formed in 1871, it was felt to be important to symbolize their whole nation with the eagle symbol (the previous incarnation, the North German Confederation, had not used an Eagle for their arms, although it was in the arms of the Kingdom of Prussia, which led the confederation), and was ordered so by Wilhelm.

To the Reich Chancellor Prince of Bismarck. Following your report of 27 June of this year I authorise: 1. that public authorities and public servants, appointed by the Emperor according to the requirements of the constitution and the laws of the German Empire, are to be called imperial; 2. that the black, one-headed, rightward-looking eagle with red beak, tongue and claws, without scepter and orb, on the breast shield the Prussian eagle, overlaid with the shield of the House of Hohenzollern, (i.e. with inescutcheon of pretence of Hohenzollern ("quarterly argent and sable")) over the same the crown in the form of the crown of Charlemagne, but with two crossing bows, may be brought into use...

Even when the empire faded into the Weimar republic, the eagle symbol continued to represent the nation.

This of course brings us to everybody's favorite group of people in history, the Nazis. It's important to remember that Hitler wanted everybody to think of his empire as legitimate, he wasn't some upstart lunatic conquering half of Europe for shits and giggles, he was taking back land that belonged to his people! Nazi Germany called themselves the Third Reich to try and suggest continuity, that they were the rightful successors of the second reich (the German Empire), and the first reich before them (the Holy Roman Empire), as suggest by the German nationalist author van den Bruck in 1923.

Circling back to the United States, the founding fathers often compared the fledgling nation to the Roman Republic; in that regard, it would be impossible for somebody to not suggest an eagle as part of the symbol of the nation, though the idea to use the uniquely American bald eagle is attributed to Continental Congress secretary Charles Thomson.