There were indeed a significant number of Jews who left Israel for elsewhere, but it's unclear if that was due to violent conflict!

First, the background: following the 1948 war and Israel's establishment, there was massive immigration into the state, primarily from the Arab and Muslim world but also from displaced persons camps set up following WWII in Europe. The immigration virtually doubled Israel's population within 3-5 years, but there was a trickle of emigration which picked up after that. Israel at that time was not just still at war and feeling the constant sense of insecurity that would stick with it for decades, but it also had economic difficulties with that absorption. For a decade or so at least, Israel had refugee camps of its own, much like those of the Palestinians at that time. However, unlike the Palestinian camps in the Arab world, Israel dismantled the camps eventually. Still, the conditions were not favorable; metal shacks in desolate areas were sometimes all that new arrivals received, until Israel began to improve its situation economically. These refugee camps, referred to as transit camps (ma'abarot in Hebrew), held over 150,000 people (in a country of less than 1.5 million people, meaning around 10% or more) in 1953. By 1963, the number was down to 15,000 or less. Around 10% of those who lived in these ma'abarot, and who came to the country following the 1948 war, ended up moving on. They left for the US when possible, but immigration law in the US was highly restrictive on that front, so many moved to places in other parts of the world, like Argentina or Canada. Israel did not make emigration easy, requiring exit permits, but one could leave if they wanted. Israel's economic situation was not great, and wars continued to occur, but they were often resounding victories, so the sense of insecurity didn't necessarily cause massive outflows. Yom Kippur created a different perspective due to the widespread perception that it was a loss in 1973, and the possibility of emigration to the US was greater because the US had removed the national-origin quotas on immigration in 1965. And indeed there was a wave of around 50,000 Israelis who did move to the US at that time, but the emigration that followed was smaller in the 1970s, around 35,000. This could have been due to any number of reasons; Israel was feeling more secure for many due to the interim agreements the US helped secure with Egypt, the eventual peace treaty which many saw as possible when Sadat visited Israel in 1977, and the economic issues of the US during this period (not to mention political issues). Of course, Israel also began to experience hyperinflation in the 1970s, so it's unclear if it was seen as any better. Ultimately it's hard to pinpoint one reason, but the numbers don't lie; the US was not a top destination for Israeli emigrants in the 1970s.

Thus, despite emigration existing, it doesn't seem to have been motivated by insecurity or ramped up due to war in large numbers. Instead, emigration to the US was rather consistent before and after the Egypt-Israel peace treaty, a surprising fact when one considers that the increased safety following such a treaty (if safety was a top concern of emigrants) should have decreased emigration, when instead emigration was roughly consistent.

So as for Jews headed to the US out of Israel, or headed out at all in fact, it doesn't seem to have been motivated by insecurity or war. Instead, it seems to have been tailored to economic benefit; the US was seen as a very attractive destination in the 1980s, so more Israelis left for the US then than the 1970s, and the 1960s were also healthier economically (arguably) than the 1970s, plus the loosened rules likely unleashed a large initial wave of emigrants.

When it came to the number who moved to Israel, it's hard to know necessarily why they moved. However, it seems unlikely to me at least that the immigrants arriving in Israel were necessarily motivated because they felt a need to stabilize the country. They may have felt a desire to help participate in the national project of a Jewish state, felt that they were more at home in it, etc. However, many simply donated money or volunteered to collect donations to send to Israel instead, particularly in the US. Instead, the biggest draw of immigration to Israel seemed to be in response to antisemitism's rise, or feelings in Jewish communities that they were in danger. From the Arab and Muslim world, this was clear particularly after 1948. In Iran specifically, a bit more of an outlier in that respect, it came particularly after the 1979 Iranian Revolution; thus, while Iran maintained a larger Jewish community than most of the Arab world, there are still far more Iranian Jews who moved to Israel than remained in Iran, and within a year of the 1979 Revolution more than 20% of the Jewish population of Iran had already left, many to Israel.

The same was true of the Soviet Union. There was an obvious "push" factor already, with the collapse of the Soviet Union creating instability that many sought to flee. However, antisemitism in the Soviet Union had already led many Soviet Jews to seek emigration, only to find it barred by the Soviet leadership. When the limits were removed, emigration to Israel exploded, resulting in up to a million (within the past 20 years, but in the early years of the 1990s was the biggest burst) Soviet Jews entering Israel, another huge number greatly enhancing its population.

So while a trickle may have desired to participate in the national project, it's hard to identify them for who they are. Some simply wanted to live among other Jews. Some viewed Israel as an attractive place to live due to the culture of the people, which has been unique in itself. Others fled antisemitism, or fled conflict in their home states that also included targeting of Jews but was also a wider conflict (i.e. Ethiopian or Yemenite Jews).

The immigration ebbed and flowed on this basis far more than based on Israel's level of security needing to be stabilized. Thus riots against Jews in Morocco coupled with Israeli encouragement of emigration out of Morocco led to a flood of Jews leaving Morocco in the 1950s, followed by the Moroccan king implementing limits on Jews leaving Morocco, a trend that ended when Morocco agreed to let Jews leave only if he got paid per Jew who was allowed to leave by Jewish organizations/Israel, and most of the remaining Jews left Morocco in the 1960s as a result. Jews were similarly already leaving communist Poland in the 1950s, but in the 1960s particularly following the adoption of antisemitic policies after the 1967 war (the Soviets had supported the Arab side), a flood of Jews left Poland, leaving it nearly empty of Jews. In Romania, economic instability and a repressive regime, as well as still-existing (but not nearly as common as in places like Poland in 1969) antisemitism led Jews to largely flee Romania in huge numbers in 1969, also in exchange for cash (prior to that point, limits were in place, but some could leave in exchange for foreign aid from Israel to Romania). Jews in the Soviet Union who could leave did so in the first wave before the 1990s, leaving in large numbers in the 1970s, particularly due to restrictions on religious Jewish life and practice as well as the continued antisemitism issue I've been describing.

I think it's far more accurate to say that for most of Israeli history, its immigration has been the result of Jews being pushed out of the states they used to be in, and simply choosing Israel because it provided the best promise of avoiding a repetition of that "push" of antisemitism or instability, with the exception of Europe and the US (which were not always available as points to leave to).