Israel is a shit-show and you won't hear me ever defend it or Zionists but these are not "white" people.
I agree with what you've said, although their true ancestry derived from their Jewish genetics is more like 30-35% with the rest being Levantine ancestry absorbed through other Eastern Mediterranean on the Greco-Roman side, post-migration.
I would say Ashkenazi Jews are similar to how Italians are white, and I would say Ashkenazi Jews are not white like how Sicilians are not white to me. In this case, Ashkenazi Jews, regardless of sticking out in some places, were also accepted by Apartheid South Africa as belonging to the white class. In America, the Jews historically enjoyed being white, and they do to this day since most of them look no darker than Italian Americans.
Although these racial classifications are different, Jews are definitely a Western people. Hands down, I would think it would be crazy to argue otherwise. The Jews are a sub-class of Western peoples. It was not always like this historically, because when they were under the Muslims, they were very MENA. And let's be real, there is a class thing. The reason why many consider Ashkenazis to be white is because of their socio-economic alignments. They're practically elites who have more in common with white elites than colored people. They assimilate easily.
In that sense, are Jews white by genetics? No. They're sort of a MENA shifting intermediate Mediterranean group. Like Cypriots. I would not consider Cypriots white, but best believe a Cypriot (Greek, not Turkish) who moves to America would mirror the rest of the clearly white people and fit in. And the interesting thing is, Turkish Cypriots who are genetically similar to the Greek Cypriots would not be white and more identify with the brown peoples. Jews are basically Greek Cypriots. I think this is the perfect analogy. Because I think Cypriots have the same type of basal MENA as Western Jews.
There are also the elements of Ashkenazi Jews living in Slavic and Germanic lands so much absorbing some genetics and many lifeways, where their last names became similar to those people ending in "-berg" or "-ski" - their European existence is still relevant despite sticking out. An African American was literally a slave for centuries in America, but those guys are a Western population, not African.
Let's take Theodore Herzl, he was a Europeanist elitist type. He saw himself as the Germans and the French, not like the Palestinians. Whenever he spoke about MENA groups, it was like a white man speaking of them at that time. They were foreign to him, and that is how Ashkenazi Jews were: Europeans. That is the reason those colonialist Zionist Brits considered Jews to be a mediating projection of their imperialism, a buffer and representation of them in the "Orient." They were like "We cannot talk to these Arabs, but the Jews can, and Jews are like us." In that dichotomy, Jews lean in with the whites and away from the Arabs.
Now you might say, but what about the Sephardic, Mizrahi, etc. Well, they're Ashkenazified. When those people came to Israel, they were dominated by the ruling elites who were European Jews, and they had to homogenize themselves to that. That is why you see Jews who are Arab (let's be frank) now suddenly act like the ones who came from Poland. There was an assimilation process where the brown Jews had to Westernize, become less Arab, and literally just divorce themselves from their own cultures for a Western European group who is indistinguishable from "modernist" white, like Ashkenazi.
I'm not going to include how awkwardly MENA and Jews too classified as "white" in America (that would never happen here in Europe) because I understand that people fighting to be "white" category kind of would have given you higher access and success chances. Like how Isaaq wanted to be classified similar to the Indians in early colonial Kenya. This would have given them privileges in a hierarchical society. And this was later used against Somalis who were actually native to the land, where the Kenyans did not give them legal status, as the Isaaqs were classified as "other." That is just an example of how Somalis, such as the clan I belong to, took advantage of a colonial system to increase their prospects. These Isaaqs were technically elites financially, so it paid off. We were new to that region, unlike the Somalis of the NFD, and the anti-Somali sentiment among other groups who wanted to suppress the Somali growth and significance and self-determination, tried to spin the false notion that Somalis there were like the Isaaq newcomers, not native like the Indians. That is just an example.
To go back, a brown Jew then is a Westerner, he is homogenized like the Ashkenazi and would enjoy white privileges as an honorary individual in white majority countries, and would not be treated like an Arab.
Historically, this was not the case. Zionism homogenized Jews. Jews were very heterogeneous. Like a Jew from Iraq had more in common and would view a Jew from Lithuania in the 1800s to be very distinct from them. One is white, the other is Arab. Two culturally distinct peoples.
We have an example of how this shift happened. In Egypt, the Jews there were practically doing well (these people were integrated with the elite class), but during the WW2 era, Zionists from Israel influenced some people there, so some Egyptian Jews became Zionists while others were anti-Zionist and sided with Palestinians, and they were very pan-Arab. Since Jews were the elites there, many of them had cosmopolitan connections with Europe. So these guys were not localized. This led to people quickly learning about trends and siding quickly. Jews in Egypt were connected to Jews from other places, like Sham. For example, in the 10th century, there was a religious institution called the Palestinian Ganonate where Jews sort of centralized a commonality through religious mediation in a sub-regional way. So I would imagine a Syrian Jew and an Egyptian Jew would be similar in their ways.
So these Jews were often elites, educated, cosmpolitan, generally more aware and engaged with European matters and at some point they saw the West growing stronger, Israel popping up with the Zionist propaganda of holyland, then some friction with the Arab locals feeling rightfully some type of way of some of the demographic who were outwardly Zionist, being favored by colonialists and also expressing Western leaning lifeways and questioning their loyalty as they clearly were very elite, Western and often were growingly Western interacting with the Ashkenazis.
The Israeli European Jews who, in clandestine ways, had come to Egypt to speak of the Zionist ideology (this is true), with that sentiment growing, added that many Jews in Egypt were somewhat integrated with Europe, stoking a growing problem. A section of them were early Egyptian nationalists who were not Zionist, but they too had interests that aligned with the other side in terms of interests and ethnic entanglement, so over time, you had a sort of quiet betrayal of Jews over Arabs, going to Israel and/or to Western nations like America. These Jews became completely Western and removed their Egyptian affinity that had conditioned them for many centuries. So a random Egyptian coming across a guy of an Egyptian Jewish background from his grandparents would not relate on that end. His parents probably would not.
That is a short and patchy example of a complex situation that shows how brown Jews became Western and now would identify with a random white man rather than an Arab, despite having grandparents who lived in a Karaite or other neighboring quarter in Cairo.
As such, these brown people became Western and in Israel, not only Western but Ashkenazi Jewish expressing. You have a handful of Karaites in Israel. Those guys pray prostrating like Muslims, but they are considered like Amish, and their youth are joining the general Ashkenazi type of Jewry, which is the dominant.