Judaism or Democracy: What Would You Choose?

In the Declaration of the Establishment of the State of Israel, signed on the eve of Israel's War of Independence, the founders of the State tiptoed around a fundamental paradox. Their attempt to evade this glaring issue, probably because they could see no ready solution to the problem, has been the root of many, if not most of the major debates in Israel and across the Jewish world.
It's worthwhile to spend time studying the document in its entirety, but I'll focus on two small sections for now.

השואה שנתחוללה על עם ישראל בזמן האחרון, בה הוכרעו לטבח מיליונים יהודים באירופה, הוכיחה מחדש בעליל את ההכרח בפתרון בעיית העם היהודי מחוסר המולדת והעצמאות על-ידי חידוש המדינה היהודית בארץ-ישראל, אשר תפתח לרווחה את שערי המולדת לכל יהודי ותעניק לעם היהודי מעמד של אומה שוות-זכויות בתוך משפחת העמים.

The catastrophe which recently befell the Jewish people - the massacre of millions of Jews in Europe - was another clear demonstration of the urgency of solving the problem of its homelessness by re-establishing in Eretz-Israel the Jewish State, which would open the gates of the homeland wide to every Jew and confer upon the Jewish people the status of a fully privileged member of the comity of nations.
Survivors of the Nazi holocaust in Europe, as well as Jews from other parts of the world, continued to migrate to Eretz-Israel, undaunted by difficulties, restrictions and dangers, and never ceased to assert their right to a life of dignity, freedom and honest toil in their national homeland.

I need to make a small but critical point here: there's a problem with the translation in the text. I didn't translate here. The Israel's Ministry of Foreign Affairs did. Notice the section that I've highlighted in bold and red: "the problem of its homelessness", which is a paraphrasing of the words, בעיית העם היהודי. While technically correct, it's missing the critical nuance of the language that the framers clearly intended. When we translate the words בעיית העם היהודי literally, we find that this phrase means "the problem of the Jewish nation." This is clearly a reference to the "Jewish Problem" articulated by Nazi Germany. See also here. As we all well know, the Nazis came up with a rather ingenious solution to the problem. I see this minor translation (or the lack of nuance) as critical, because it highlights how strong a role the Holocaust played in the creation of the State of Israel, not only in the minds of the members of the international community, but also in the minds of the founders of the Jewish State.
With this is mind, we can begin to get a sense of the urgency of creating a Jewish State. Jews were slaughtered by the millions, with nowhere to run for refuge. The founders of the State declared openly that this would never happen again. Israel would stand ready to accept any Jew running from persecution. It would be a haven for Jews fleeing from the rampages of antisemitism. It would do so, by definition, by creating itself as a Jewish State: by the Jews, and for the Jews.
But then, later on in the document, the framers make another important statement:

THE STATE OF ISRAEL will be open for Jewish immigration and for the Ingathering of the Exiles; it will foster the development of the country for the benefit of all its inhabitants; it will be based on freedom, justice and peace as envisaged by the prophets of Israel; it will ensure complete equality of social and political rights to all its inhabitants irrespective of religion, race or sex; it will guarantee freedom of religion, conscience, language, education and culture; it will safeguard the Holy Places of all religions; and it will be faithful to the principles of the Charter of the United Nations.

In other words, Israel would be a democracy, granting full rights to all of its citizens. Sounds great. But there's only problem. How do you ensure that the State remains Jewish if it truly adheres to the principles of democracy fully? If you grant "complete equality" to all citizens, what do you do if a minority grows to the point that it threatens to overtake the Jewish minority?
Let's leave the territories of out the equation. What do you do when you see secular Israelis fleeing the country, while the Israeli Arab population explodes? Would the State of Israel still be a Jewish State if its Arab majority voted for an Arab Prime Minister? How then do you guarantee that the State remains both Jewish, and a democracy?
In a word, you can't.
Until now, we've been avoiding this internal contradiction. We haven't had to confront the truth: a Jewish State and true democratic values might not be completely compatible. Something has to give.
So you've got to choose. Which are you willing to sacrifice? Are you willing to risk Israel's status as a Jewish State for the sake of the principles of democracy, equality and fairness, hallowed and sacred and important values? Or, will you sacrifice that equality for all to ensure that Israel remains a Jewish country?
That's the choice. Which would you choose?