So more, and less, details have come out regarding Rav Kanievsky's advice about expelling the students who got themselves licenses, as discussed here.
The new article, at YNet, clarified that it was against the particular yeshiva's policies from the outset, and the bachurim knew this, and got driver's ed and then licenses anyway. The yeshiva is now named. Is it for Israelis or for Americans? I am not familiar with the particular yeshiva, but if one of the students was doing to help his disabled father, who would be in Israel, I would guess that it indeed is for Israelis. The number of students expelled was downgraded from 10 to 4. And they were expelled conditionally -- they get to come back as soon as they cancel their licenses. The implication is that the one with the disabled father was only advised to cancel his license, but that he can decide, and in the meantime (and regardless), he can stay in yeshiva. The YNet article also does not mention Rav Kanievsky's name in this regard, perhaps because the sources to the reporter refused to mention Rav Kanievsky's name.
I still do not understand the advice to the bachur, which appears to be that one is not considered a "ben Torah" if he has a license, so one is advised to cancel it and make one's disabled father remain stuck in the house, unable to get places. And that that behavior would make one a ben Torah. (The general attitude regarding learning to drive, while I disagree with, I would chalk up to cultural differences. Let me add that this is nothing to do with defensiveness -- I don't have a license myself, but that is my flaw, not to my credit.)