re "Misnagdic.... Jews are forbidden" etc
How is this news? The sign that a bunch of top haredi rabbis put up at the Mount after the 67 war is still posted there saying it's forbidden. It's an old argument. There have always been some rabbis who argued it was not forbidden. While many haredi rabbis argued it is forbidden. Funny, in the article there is no reasoning given at all. Rav Eliyashiv is only repeating the precendent set by his predecessors. He even says in the article he's had this opinion for a long time.
What would be Rav Kahane's and his son's Halachic view about ascending the Temple Mount today?
What was it in their lifetime? There is no reason why it would have changed. This is an old issue. I heard a firsthand account of someone who said that he witnessed a time when Rav Kahane went up to Temple Mount, but I don't know if that was reliable account. Maybe someone in the forum knows or maybe Chaim knows.
Or do current exegencies and the Chillul/Kiddush Hashem of the Jewish right to ascend the Har Habayis today transcend even Halachah?!
What halacha exactly? Can you point out which one?
The majority Haredi position has always been that it is 'assur' to go onto Temple mount. That explains the sign that is posted up there. I have not seen a good explanation for the reasoning behind that, in light of the rather good explanations given by the opposing side that argues it is permitted, nor does the article you cited give any reasoning behind Rabbi Eliyashiv's opinion. They only quoted the soundbite that fits their agenda. But there have always been other Torah authorities, some National religious rabbis, and some haredi rabbis, who have differed in their opinion and have given good reasons that it is NOT forbidden. So why are you so quick to say it's "the halacha" ? Let's see if you can point out in what way it is the halacha or what halacha you are relating it to. You can't necessarily guess at what Rav Eliyashiv's reasoning is, but since you are so convinced, I assume you have some other information not available to us. So please share.