That's precisely what Zionism is and has been. Jews seeing themselves as belonging to a nation grew to be a more common opinion over time. Part of the difficulty of this was the spread out nature of the Jewish diaspora. European Jews had a very different, more intimate experience with Zionism as a national identity than Middle Eastern ones in the 1800s and even much of the 1900s.

Zionism as an idea is old. Biblical texts refer to Jews as a nation. But the belief that Zionism should mean a state for Jews was difficult to imagine. Where such a state would or could be was disputed, and whether it was right to seek such a state given the Jewish perspective on the messiah's arrival having a specific order (messiah first, state later) was contentious. Even so, Zionism saw significant writings and flavors grow in popularity mostly among European Jews with the writings of Moses Hess in the 1860s, and Pinsker in the 1880s, for example. By the 1880s, some Zionist Jews were leaving parts of Eastern Europe and Russia to try and move to what became Israel, and create new Jewish communities there. Herzl, the founder of political Zionism as a movement more or less, wrote Der Judenstaat by 1896, a pamphlet that held significant message for Zionist advocates on the methods, plans, and ideas behind a Jewish state.

Were Jews unified? Not for a long, long time. Middle Eastern Jews forced to or fleeing to Israel following the 1948 war were not necessarily strong Zionists, for a long time. It was a largely European-dominated phenomenon, much in the same way nation-states was largely a European-dominated phenomenon to begin with, as far as I've understood it. But Jews did see themselves as a nation deserving of a unified state over time, and the movement towards that did begin in the 1800s. Der Judenstaat said, in fact:

No one can deny the gravity of the situation of the Jews. Wherever they live in perceptible numbers, they are more or less persecuted. Their equality before the law, granted by statute, has become practically a dead letter. They are debarred from filling even moderately high positions, either in the army, or in any public or private capacity. And attempts are made to thrust them out of business also: "Don't buy from Jews!"

And it noted then that:

I referred previously to our "assimilation". I do not for a moment wish to imply that I desire such an end. Our national character is too historically famous, and, in spite of every degradation, too fine to make its annihilation desirable. We might perhaps be able to merge ourselves entirely into surrounding races, if these were to leave us in peace for a period of two generations. But they will not [92]leave us in peace. For a little period they manage to tolerate us, and then their hostility breaks out again and again. The world is provoked somehow by our prosperity, because it has for many centuries been accustomed to consider us as the most contemptible among the poverty-stricken. In its ignorance and narrowness of heart, it fails to observe that prosperity weakens our Judaism and extinguishes our peculiarities. It is only pressure that forces us back to the parent stem; it is only hatred encompassing us that makes us strangers once more.

And he resolved, from this point, that:

Thus, whether we like it or not, we are now, and shall henceforth remain, a historic group with unmistakable characteristics common to us all.

We are one people—our enemies have made us one without our consent, as repeatedly happens in history. Distress binds us together, and, thus united, we suddenly discover our strength. Yes, we are strong enough to form a State, and, indeed, a model State. We possess all human and material resources necessary for the purpose.

For Herzl, he saw a pattern. He viewed it in the way that most bigotry is viewed: as unwinnable. You cannot satisfy those with irrational hatred of Jews, he argued.

When Jews were shunned, they formed their own communities.

When they formed communities, they were attacked for being insular.

So they tried to assimilate. But this failed, and Jews remained marked and were still attacked (and this is 40-50 years before the Holocaust, mind you).

So now, they must give up on assimilation and assert their right to equal statehood among the nations, to give Jews an equal standing in the world and a place of their own in which they can live without fear of prejudice or the tides of anti-Semitism in government.

So yes, Herzl saw a Jewish nation, as did many others, even in the earlier parts of the 1800s. But it wasn't immediate, and the idea spread over time.