Parashat Emor: Cohanim and Kivrei Tzaddikim (the graves of the righteous)

In this week's Parashah – Parashat Emor, we learn that it is forbidden for cohanim to become impure by coming in contact with the dead, except in the case of close relatives. A small minority of authorities hold that the Kivrei Tzaddikim do not transmit impurity (see Pitchei Teshuvah Yoreh Deah 372:2).
Our Rabbi, Ha-Rav Tzvi Yehudah, told the following story about Maran Ha-Rav Kook (Le-Shelosha Ba-Elul vol.1 #76), that during his travels to oversee Jewish matters in the Galil, he did not visit Kivrei Tzadkim when he was in Tzefat, because he was a Cohain.
Ha-Rav Mordechai Eliyahu wrote (Parashah Sheet "Kol Tzofa’ich" #279), "In his time, I told Ha-Rav Ha-Gaon Tzvi Yehudah Ha-Cohain Kook, peace be upon him, that it is written in the book ‘Kuntres Yechi’eli’ that it is permissible for cohanim to enter Kever Rachel. He asked me: what do they say there? I said that they read the verses about our mother Rachel there. He travelled there, but only went as far as the door. When he returned, I asked him: why didn’t you enter? He answered: My father did not enter, therefore I did not enter."
In the book Sichot Ha-Rav Tzvi Yehudah - Eretz Yisrael (edited by Rav Shlomo Aviner, p. 142 note 1), it relates that after the Six-Day War, the students of our Rabbi organized a trip to the liberated areas in the Shomron. One of the places they visted was Kever Yosef. The students entered inside, but our Rabbi remained outside, because he was a cohain.
And on Maran Ha-Rav Kook's yahrtzeit, our Rabbi, Ha-Rav Tzvi Yehudah, would visit his grave on the Mt. of Olives, but would stand at a distance since he was a cohain (Be-Derech Ha-Torah Ha-Goelet p. 170).
In Iturei Cohanim (Sivan 5766 #261), Rav Aviner was asked, is it permissible for a cohain to enter Ma’arat Ha-Machpelah? He answered that there is a dispute, but Maran Ha-Rav Kook did not enter. In Shut She’eilat Shlomo (vol. 3 #329), Rav Aviner also writes that although there are authorities who permit cohanim to enter "Kivrei Tzaddikim," since the righteous are called "living even in their death," the accepted halachah is that it is forbidden. There are also authorities who allow cohanim to visit Maarat Ha-Machpelah and Kever Rachel, because they were built in a way that the cohanim would not become impure; but the acceptable halachah for this is also that it is forbidden. Therefore, we say that cohanim should not enter "Kivrei Tzaddikim," but we can defend the practice of those who act in this way, especially entering Maarat Ha-Machpelah and Kever Rachel (note: Rav Aviner – who is also a cohain – has not and does not visit any of the "Kivrei Tzaddikim").