This is a good question. It shows to me that you are trying to think deeply about the Scripture. My answer is both yes and no. Let me explain. Your premise is that the destruction of the Southern Kingdom and the exile of the Jews, after which they never had a political king ruling on a throne in Jerusalem caused them to have a messianic expectation. I will agree with this premise. This proposal is both reasonable and supported by evidence. It is reasonable to assume that the Jews looked for a different kind of king when they no longer had a political king on the throne in Jerusalem. Given the promise to David in 2 Sam 7:12-16, the Jews expected to always have a king on the throne. Yet, after the death of Zedekiah, there was no longer a king in the political sense. It is not surprising that the Jews began to think of a different kind of king. They found this idea already in their scriptures. I could give a LOT of examples, but one scripture which comes to mind is Zechariah 9:9-10, in which a different sort of king is described—a lowly king, riding on a donkey, not on a horse. Yes, I will agree with your premise that the political situation after the captivity in Babylon and the restoration under Cyrus increased the Jew’s awareness of the biblical truth that God was to send a Messiah. Yes, the idea of a Messiah “developed” after the exile.
On the other hand, whether I completely agree with your proposal will depend on how you are using the word “develop.” What I mean is this. Yes, the exile and return caused the Jews to be more aware of the idea of a different kind of king—the Messiah. But it is not that there was no idea of a Messiah before the exile. If “develop” in your last sentence means that it originated with the exile, then I will disagree. There were plenty of passages which preceded the exile which told the Jews about a Messiah. There are messianic passages such as Psalm 110:1, Isaiah 53:1-12, MIcah 5:2, and Deuteronomy 18:17-20, which strongly point to a Messiah before the exile.
What we do not have is a lot of evidence from Jewish literature outside the Bible from before the exile to tell us the degree to which they understood that a Messiah was coming. Besides, many of the clearly messianic passages, such as Daniel 9:24-25, Zechariah 9:9-10 and others are from the exile or post-exile. We simply do not have the evidence to tell us how strong the messianic expectation was before the exile to give a solid answer to how much the messianic expectation increased among the Jews after the exile.
So, I will agree with you that, yes, the idea of the messiah did develop after the exile, as the Jew found themselves out of political power. This caused them to realize that the kingship of David would take a new and greater form in the future. The Essenes had a strong sense of the role of the Messiah, as did many in mainstream Judaism. However, the idea of the Messiah did not begin with the exile. The Jew’s awareness increased, but it did not begin with the exile.