Such an interesting question! I may be shockingly unqualified to answer it from one perspective as a Jew who really did not grow up hearing it around much; I grew up in an Orthodox Jewish community in which other Jewish songs were a lot more popular, and really became most familiar with it in high school when my music teacher, who was a music director at a Reform temple, assigned it to me as an exercise. But of course I do know that it's a very popular one, even if I didn't know all of the words until about half an hour ago. (Of course, now after researching it it's come out of my ears lol.)

You're sort of asking two different questions- how did Hava Nagila become a symbol of Jewishness FOR JEWS, the kind of song played at every wedding and bar/bat mitzvah, and how did it become the kind of song that Bruce Springsteen covered. Dividing the question is important, because it's really that the answer to the first question answers the second, in a way.

The answer to the first question is that it became a popular Zionist song, as you observed. Those aren't actually its origins; the tune (not lyrics) are originally a chassidic tune, from the chassidic court of Sadigura (Sadhora) in Bukovina, then part of the Austro-Hungarian Empire. The Sadigura court was established by the son of Rabbi Yisrael Friedman of the long-standing and highly esteemed chassidic court of Ruzhin in the Russian Empire, who had fled to Sadigura to avoid persecution. It was a form of niggun, or melody used during religious celebrations which often contained no words, that could be slow and meditative or lively and joyful. [EDITED: ...and I just read your OP and realized you already knew that lol] I've heard some variations on it still used in chassidic contexts.

It became Hava Nagila, words and all, because of Abraham Zvi Idelsohn, who was, at the time of WWI, a cantor and scholar of Jewish music in Jerusalem who sought both to record old Jewish tunes and to establish a new Jewish music that relied on the Jewish themes of the past rather than being totally indebted to modern music. It's unclear how he came across the tune; he claimed that he heard it from a chassid in the Sadigura synagogue in 1915, but documentary evidence indicates that he may have heard it on a visit to Europe a couple of years earlier. It's also unclear how much credit he can take, as there was a dispute between him (he claimed that he wrote the lyrics and arranged the tune) and one of his students, Moshe Nathanson, who said that he had actually written it for Idelsohn as part of a class assignment. Either way, the song came at the right time; with the Balfour Declaration, the Jews of Palestine felt newly hopeful about their status in the Holy Land and their ability to establish a Jewish, Zionist culture, and Hava Nagila, with its upbeat tune and joyous lyrics (which were, in accordance with Idelsohn's dictum that the new Jewish music should be traditionally sourced, based on Psalms), was the song for the moment. The song was perfect for use as a hora, a new popular dance in the yishuv, or Jewish settlements of Palestine. It became extremely popular in Palestine, and continued to be so in the State of Israel.

It was because of its joyous nature and its connection to Palestine/Israel that Hava Nagila arrived in the US. It happened almost immediately, as the tight-knit nature of the Jewish community led the song to spread worldwide. As the Zionist movement became more and more popular in the Jewish diaspora, especially with the onset of Nazi persecutions of Jews in Germany and then the Holocaust, the culture of Zionism became a pervasive part of Jewish life in the US. It was at this point, in the 20s-40s, that the song became a popular one to be sung at weddings, bar/bat mitzvahs, and other Jewish events, especially as the hora was such a popular dance at these celebrations.

And it was, perhaps, because of how popular the song was at Jewish events that it became so popular in the world at large. For example, Glen Campbell, who released a version of Hava Nagila in the late 60s, originally learned the song so that he would be marketable to play Jewish events. Either way, beginning in the 1950s, the song began to leak into the mainstream at a time when many artists were recording cross-cultural hits. In 1951, a mambo adaptation was produced (or so it's claimed- personally, I can't really hear it) by the Cuban mambo band Machito y sus Afrocubanos; not long after, in probably the most important development in making the song as popular as it became, Harry Belafonte found the song, liked it and the universality and hopefulness of its lyrics ("come let us sing and rejoice, arise brothers with a joyous heart"), and put out his version. In the documentary "Hava Nagila (The Movie)," Belafonte claimed that it was one of his two most popular songs requested. After this, as you note, it became a song covered by tens of performers completely outside a Jewish context, many of whom were not Jewish themselves; the singer Connie Francis, who regularly ended her shows with Hava Nagila, would respond when asked if she was Jewish that she was "10 percent Jewish — on my manager’s side!’’

And then, as with so many things that become so popular in a group that they end up blowing up outside it and then making their way back in... the song in many ways became over-saturated. Bob Dylan did a parody of it, though I can't find it online anywhere; that said, he then went on to play it totally straight at a Chabad telethon. It in many ways became a symbol of schlocky Jewish music, or even schlocky Judaism, to the extent that the Jewish communities that had loved it for so long got frustrated that it had turned into such a joke. Nonetheless, it has remained a beloved classic for many who see it as nostalgic of their traditions. And, in fact, there began an attempt to, so to speak, "reclaim" Hava Nagila, with a notable recent attempt being Aly Raisman, the Olympic gold medal gymnast, using the tune for her floor routine at the London 2012 games.