Which was more important to the development of Jewish religion, culture, identity, etc.: the mythical exile in Egypt, or the actual exile in Babylon?
This is a loaded question, but I think it boils down to the question of whether subjects of the Kingdom of Judah prior to the Babylonian sack were recognizably "Jews" or whether "Judaism" as a religion only developed in the aftermath of that event. Is this more or less what you were trying to ask? I'll come back to this one later.
The heart of my question was, How does the importance that the Babylonian Exile had on the development of modern Judaism compare with the importance of the Exodus mythos on the development of Judaism in its embryonic stage, and which one was more influential? My hope in asking this question was actually that you would talk about how Judaism developed, which is a topic that's captured my interest lately.
OK, let's take a shot at this. First off, I'd say that this is an impossible question much in the same way as asking if the Trojan War or the Peloponnesian War was more significant to ancient Greek development: you cannot just contrast two events that are separated by over half a millennium, one of which is a deep-seated cultural icon important to the formation of the mythos of the people and the other an (albeit spottily recorded) historical event. I'm choosing to interpret this question as A. How important is the Exodus story to Judaism, and B. how fully-formed was Judaism prior to the Babylonian Exile vs the extent to which it was created by that epochal event.
A is probably the easiest one to dispense with. If Moses is a fully legendary figure, which seems very likely, he is probably the single most influential fully legendary figure ever, rivaling King Numa and the Yellow Emperor and surpassing figures like King Arthur and Lycurgus. The Exodus narrative is critical to the way that the early Israelites saw themselves and their place in the world: as exiled refugees and therefore outsiders in the land of Canaan, as a people who had formed a divine contract with a God far more powerful than the gods of the locals (let's sidestep the issue of whether the earliest Israelites were monotheists for now and focus on that they saw Yahweh as the all-powerful creator of the universe and therefore far superior to any other god that may or may not exist), as a people who had been plucked from the lowest ranks of society to a grand divine mission. One could easily make the case that, while the Exodus myth faded from the consciousness a bit and was revived by Ezra etc. after the Babylonian Exile as a precedent for that event, the story was still one everyone was familiar with: a people exiled from the land of Egypt are given a set of laws and purpose by their relationship with their God, have violated that relationship, and met with calamity at the hands of the Assyrians and later the Babylonians. The famous story of Ezra reading the Torah to the recently-returned first wave of Jews back in Jerusalem is a story of the Jews seeing how their ancestors had ignored these rules and disaster had struck. Would this generation risk the same offenses?
For B. I'd like to address terminology a bit. There is a difference between the word "Jew" and the word "Israelite," despite their frequent conflation. The Israelites refer to all twelve tribes at first and later to the inhabitants of the Northern Kingdom of Israel and its ten tribes. The Jews are the heirs to the Judahites, or inhabitants of the Southern Kingdom of Judah, after the other tribes seceded from rule by the heirs of David in the days of Rehoboam and Jeroboam, according to the Biblical narrative. Whether the United Monarchy itself was a myth or not, the existence of the the two rival kingdoms is real enough, and until the wars with Assyria, the word Israelite is useful because the Northern Kingdom has radically different views on what the religious practices of the cult of Yahweh entails than their southern cousins.
There is a famous incident in the Second Book of Kings where King Josiah's men tear down an interior wall in the Temple during a renovation project and uncover an ancient "scroll of the law" containing various divine decrees that the Kingdom of Judah had never carried out. There's a long-standing and rather convenient scholarly interpretation that that scroll was the Book of Deuteronomy, potentially forged by those scholars in the context of the crises of the late 7th century BCE. If so, it reflects just how important the name of Moses was to the Judahite clergy: a forgery in the name of the grand lawgiver would give them all the excuses they needed to implement a program of radical monotheism, iconoclasm and destruction of pagan shrines, and the consolidation of all legitimate religious practice in the Temple of Jerusalem. Of course, when the Northern Kingdom of Israel had existed, it pointed to the many designated holy spots in its territory (most famously the shrine of Beth El, where Jacob had had his vision of the ladder) as equally legitimate places for worship and sacrifice to God to prevent pilgrims from having to leave the country, but now the Judahite elite could firmly denounce any shrine to God that was not the Temple in Jerusalem as idolatrous and centralize legitimate worship in one Temple, a move that would prove dangerous indeed when Nebuchadnezzar razed the Temple a few decades later.
Pre-Exilic Israelite and Judahite faith had been marked by a strong rivalry between priests who claimed religious authority through their hereditary roles and rituals, and "prophetic" figures who claimed to circumvent all of the pomp and circumstance by speaking directly to the divine. From the moment Isaiah had his vision of leaving the earthly Temple to enter its divine doppelganger, seeing God in all his splendor with the heavenly host crying "Holy Holy Holy is the Lord of Hosts! The whole of the Earth is his glory!" the priests had a powerful rival message. Isaiah's radical egalitarian message is deeply at odds with a hierarchical religion: in light of the recently passed Day of Atonement, a fast day in Judaism, this passage is always appropriate.
3 ‘Why have we fasted,’ they say,
‘and you have not seen it?
Why have we humbled ourselves,
and you have not noticed?’
“Yet on the day of your fasting, you do as you please
and exploit all your workers.
4 Your fasting ends in quarreling and strife,
and in striking each other with wicked fists.
You cannot fast as you do today
and expect your voice to be heard on high.
5 Is this the kind of fast I have chosen,
only a day for people to humble themselves?
Is it only for bowing one’s head like a reed
and for lying in sackcloth and ashes?
Is that what you call a fast,
a day acceptable to the Lord?
6 “Is not this the kind of fasting I have chosen:
to loose the chains of injustice
and untie the cords of the yoke,
to set the oppressed free
and break every yoke?
7 Is it not to share your food with the hungry
and to provide the poor wanderer with shelter—
when you see the naked, to clothe them,
and not to turn away from your own flesh and blood?
Figures like Isaiah pointed to the fall of the Northern Kingdom of Israel to the Assyrians as punishment for its disloyalty to God and threatens Judah with the same fate if it continues its immoral politics.
5 “Woe to the Assyrian, the rod of my anger,
in whose hand is the club of my wrath!
6 I send him against a godless nation,
I dispatch him against a people who anger me,
to seize loot and snatch plunder,
and to trample them down like mud in the streets.
7 But this is not what he intends,
this is not what he has in mind;
his purpose is to destroy,
to put an end to many nations.
8 ‘Are not my commanders all kings?’ he says.
9 ‘Has not Kalno fared like Carchemish?
Is not Hamath like Arpad,
and Samaria like Damascus?
10 As my hand seized the kingdoms of the idols,
kingdoms whose images excelled those of Jerusalem and Samaria—
11 shall I not deal with Jerusalem and her images
as I dealt with Samaria and her idols?’”
Of course, after the fall of the Northern Kingdom of Israel, the Southern Kingdom of Judah became an autonomous, humbled vassal of the Assyrian Empire. King Hezekiah sent the golden doors of the Temple to the Assyrians as a surrender tribute. Isaiah's successors, especially Jeremiah, bear a similar message in the time of the last generation of the Kingdom of Judah. Jeremiah reminds the Judahites again and again that it's not obedience to rituals God wants, but true good deeds and faith, and the faithless hard-hearted indulgence of the Judahites are leading to imminent destruction.
In short, already before the Exile you have a strand of people (Isaiah and his successors) reinterpreting God from being a tribal deity who will win wars and vanquish his people's enemies, to a universal deity who has a contract with one specific people but is willing to aid its enemies in smashing them if they refuse to hold up their end of the contract. This shift of God from God of the Israelites to universal deity with a special contract with the Jews, a God who has an emphasis on righteous conduct and care for the impoverished rather than solely interested in rituals, is well underway in pre-Exilic Judaism. The Exile forced the issue: there is no longer a Temple to sacrifice in or to worship God in the proper way that Deuteronomy suggested. How can one still be a follower of Yahweh in the city of Marduk? The messages of the prophets, both radical monotheism and an emphasis on conduct rather than sacrifice and prayer, helped provide an answer to that question.