I'd like to talk about the historical comparisons made, but first I'd like to talk about the reason why the vast, vast majority of Jews did not consider Ben-Gurion to be the messiah, and why so many did not believe he fulfilled the standards that religious authorities looked for. Many of those standards are the same ones that Jewish theologians use as a baseline for not believing in Jesus as Messiah either, such as the prediction that all the world will follow the God of Israel, for example.

Among the various requirements laid out for how a Messiah is to be identified, which Ben-Gurion did not meet, perhaps the most important and relevant to your question is the ingathering of the exiles. You see, Jewish eschatology does not predict that the Messiah will gather some of the Jews in the land of Israel, but rather that they will gather all of them there. While Isaiah 11:12 does not say the word "all", it has commonly been interpreted that way, based on the way it is written.

This is among the many other standards that Ben-Gurion does not satisfy, among them the requirement that he be exalted by the entire world (Ben-Gurion was decidedly not), that at that time the world will beat their swords into plowshares (Ben-Gurion was decisively in favor of Israel developing nuclear weapons, not the other way around), and that he will have a spirit of devotion to God and understand justice innately, which is rather ironic for Ben-Gurion, who many suspected was a secret atheist (as far as anyone can tell, he was not an atheist, but he was certainly no devoted scholar of Jewish thought or follower of its tenets).

For these and many more reasons, Jewish theologians largely eschewed the idea that Ben-Gurion could be considered the Messiah. Indeed, some eschewed the idea even that Israel should even be founded, given that Ben-Gurion was decidedly not the Messiah, and they believed that to preempt the Messiah's task of gathering Jews in the land of Israel was wrong. There are a variety of ultra-Orthodox Jews, such as the Satmar Hasidim, who are anti-Zionist and believe that Israel's creation was wrong. Rabbi Joel Teitelbaum, a prominent Satmar Hasid, called Zionism and the Jewish state heretical, particularly given its secular founders and founding ideas, and argued that the Talmud forbade creation of the state. Of course, Teitelbaum was argued with by other religious figures, particularly after the Holocaust and the founding of the State of Israel, both of which created a sea change in Jewish thought.

Ben-Gurion never tried to pass himself off as Messiah, though he was often accused of doing so thanks to his use of messianic language. Ben-Gurion attempted to use that language, most now agree, to provide a unifying factor that could bring Zionism as an idea to the various Jewish groups of the Middle East and Europe coalescing into a single-state, and give them a unifying idea. Israel was thus made from a unit redeeming and protecting the Jewish people from a second Holocaust (a theme that was not reinforced for many Israelis until the Eichmann trial) that was harder for Middle Eastern Jews to relate to, into a state that was a messianic vision, with the utopian aspects of messianic thought that came with it, such as a devotion to freedom for Jews to be Jews, a unique state and status, etc.. Ben-Gurion himself never claimed to be a Messianic figure, however, because that effectively defeats the purpose and would also go against Jewish predictions of a Messiah. While comparisons were thus often made, they never arose a serious movement around heralding Ben-Gurion as a Messiah; he simply did not suit the criteria, and the world of the day surely didn't either. While some Jewish thinkers believe that in every generation there is a person born who could be the Messiah, the conditions must be met for them to fulfill that role, and Ben-Gurion's situation and surroundings, as well as his personal characteristics, were far from fulfilling of it. There was debate over whether the creation of Israel would herald the start of the Messianic Age, where the Messiah would arise, but Ben-Gurion was not believed to be said Messiah, nor did that happen; as you may have guessed, the world has not beaten its swords into plowshares. It might have been reasonable to argue, as some did, that the creation of Israel in turmoil was not itself a contravention of the Messianic Age's characteristics; after all, some argue that for 7 years after the start of the Messianic Age, there will be a refining of the Jewish people in preparation of the Messiah's arrival. This, of course, does not line up with history, and does not line up with Messianic predictions in Jewish eschatology.

In short, to argue Ben-Gurion was the Messiah was too far a stretch for the vast majority of Jews, if not all Jews. Merely accepting the state of Israel was difficult for the most religious (but also, least populous) Jews, many of whom remained reluctant to accept Israel even while accepting benefits from its existence as time went on. Ben-Gurion failed to meet many of the criteria for a Messiah, and the world never mirrored the predictions laid out in Jewish thought. In fact, the majority of those who argued that Ben-Gurion was a messiah were in fact arguing in line with Messianic Christianity that Ben-Gurion's actions fulfilled one of the precursors to the return of Jesus, which is a theme that evangelical Christianity has also pushed in its staunch defense of Israel's existence and of Jews gathering there (as a precondition to Jesus's return, in Christian thought).

The Messiah is meant to do a number of things. Wikipedia has a decent list. One of the most important is rebuilding the Temple, something which the secular circle of early Zionists (including Ben Gurion) had no interest in. Another bigger issue was that the Messiah is meant to, you know, end death and war and all that sort of thing.

No one thought Ben Gurion was the Messiah. And most people don't even understand Ben Gurion as gathering the Jews in Israel--it was Herzl's vision, with many of the immigrants coming to Israel before Ben Gurion rose to power. Ben Gurion's public reputation is as the (or a) founder of the state of Israel, he's not really a gatherer of the scattered diaspora.

Traditionally, religious leaders were skeptical of Zionism, which was often associated with socialism (Labor Zionism). This continues to the present day with many Hasidic dynasties (most notably Satmar) being anti-Zionist and most of the Haredi ("Ultra-Orthodox") Jews simply adopting the more moderate "non-Zionist" stance. But before the Holocaust, a higher proportion were explicitly anti-Zionist--there were, among other issues, serious doubts about whether it was licit to travel en masse to the Land of Israel. See the Wikipedia page for Haredim and Zionism for more. At the time, in the first half of the 20th century, Haredim and their organizations like Agudath Yisroel, etc. tended to be pretty explicitly anti-Zionist.

Relatively few religious leaders supported the Zionist vision before around 1967. One of the notable people is the Proto-Zionist Zvi Hirsch Kalischer but I think if you compare his vision to Ben Gurion, Hertzel, et. al.'s you can see clear differences. Rabbi Kalischer, for instance, was deeply focused on thinking about what sacrifices could be done in the absence of the Messiah.

Another important figure was Rav Kook. His view was that these secular Zionists were unknowingly instruments of God's will. He was the first Chief Ashkenazi Rabbi of Israel, but his views weren't particularly influential while he was alive. However, after the stunning victories of the 1967 Six Day War, which not only defeated all of Israel's enemies in one blow but also captured all of Jerusalem putting holy sites like the Western Wall under Jewish control for the first time in about two millennia. This led to a bit of messianic fervor in Israel, and a theology where the secular Zionists were tools of God, the founding of the State of Israel was the "wellspring" of the Messianic Age, and that aggressively settling the Land would hasten the imminent Messianic Age. These beliefs are important to the "Dati Leumi" segment of the Israeli population, who are very roughly the equivalent of Modern Orthodoxy and also make up very roughly 1/4 of the Israeli Jewish population. The assassin of Yitzak Rabin, the Prime Minister who was promising "Land for Peace" and shepherded the Oslo Accords and other peace deals, was done by radical member of this tendency. Likewise, Dati Leumi make up a large proportion of settlers in the West Bank, especially in the more outlying settlements, because many worry that giving up land to a Palestinian state would literally delay the coming of the Messiah.

So the state of Israel and Zionism is tied up with a theology of the Messianic Age, but no one associated with this tendency is associated with actually being a Messiah candidate. The person most associated with being a Messiah candidate in the 20th century was the last Rebbe of the Chabad-Lubavitch Hasidic sect, Menachem Mendel Schneerson. While he never definitively declared that he thought he was the Messiah or even that he was a candidate for being the Messiah, a large section of his followers unabashedly did and some even still do. To give you a sense, he's the only significant figure from the 20th or 21st centuries on Wikipedia's list of Jewish Messiah claimants. It's a very different model of what it means to "gather" all of Israel. It is often thought that messianic signs would happen miraculously--for instance, many believe that the Third Temple will descend fully built from heaven, unlike the first two Temples which were built by specific political leaders. Also, going back further, it's worth noting the only 19th century candidates only had followers in the very isolated Jewish community of Yemen. Messiah candidates were once more common, but I think experiencing the twin failures of Sabbatai Zevi (converted to Islam) in the 17th century and Jacob Frank (converted to Christianity) in the 18th century have largely chastened the Jewish community away from messianic candidates (though of course, I think both the post-1967 settler movement and Chabad-Lubavitch show there's still room for messianic expectation). Even before Frank and Zevi, there were only a handful of Jewish messiah claimants who appear in history. It's far from a common thing. Their profiles have generally been mystical religious leaders, rather than secular political leaders, particularly after the emergence of Kabalah in the 12th century.