Joseph, Torah is not a yerusha but a morasha (see Yerushalmi Bava Basra 8:2 that morasha is a weaker form of inheritance). Many sons of talmidei chachamim have been ballei battim. Rav Moshe Soloveichik had one son who was a Chemsitry professor in YU and one of Rav Mordechai Eliahu’ssons is a frum lawyer.
As for Rav Schach, I think that he was actually a non-Zionist in that he did not accept the philosophical underpinnings of religious Zionism but did not oppose Israel’s existence as a state.. An anti-Zionist would not have called for people to vote in Israeli elections, much less found a political party.
ZD, when Rav Shaar Yishuv Cohen zatsal came to a wedding in uniform dureing the War of Independence some nudniks complained about the way he was dressed. Rav Tzvi Yehuda asked why the dress of paritzim was better.