>>>What Rav shachs understanding of the differences were is beyond my understanding; i have emunas chachamim that he saw a significant change which permitted joining the government.
It is inaccurate to ascribe the change exclusively to Rav Schach. (see below) It was the unanimous decision of the Moetzes to join Begin’s coalition in 1977. And due to change in status quo they were making they asked the American Moetzes for outside approval first. Which was quickly give to them. The reason for the change was simple. They realized that they were dealing with a different situation. It shouldn’t be hard to find a Jewish Observer online from that period that will explain it to you further.
According to Menachem Porush, Rav Schach was actually from the more resistant members of the Moetzes to joining Begin’s coalition. He insisted he would only do so if there was no chance of Shimon Peres making a coalition anyway. Because Shimon Peres had consistently opposed drfting Yeshiva Bochurim over the years and Rav Schach felt the Charieidi world owed him hakoras hatov
It’s also inaccurate to claim that the Charidie world did not take cabinet position until recently. Reb Itche Meri Levin was the minister of social welfare in 1948-1949.