Leaving Israel to visit "Kivrei Tzadikim" (the graves of the righteous)

Q: Is it permissible to leave Eretz Yisrael to visit "Kivrei Tzadikim"?
A: Although the Rambam (Hilchot Melachim 5:9) ruled that it is forbidden to leave Israel except temporarily to learn Torah, to find a wife or to be saved from non-Jews, the Sedei Chemed (Ma'arechet Eretz Yisrael #1) quotes various authorities who allow leaving Israel to visit "Kivrei Tzadikim". Maran Ha-Rav Kook ruled that we do not leave Israel to visit the graves of Tzadikim and he wrote "are we without graves in the Land of Israel that you travel to the Exile?!" Isn't it enough to visit our Forefathers and Forefathers in Ma'arat Ha-Machpelah in Hevron (Shut Mishpat Cohain #147)?
Tosafot (Avodah Zarah 13a) ruled that one may leave Israel for any mitzvah, but we must clarify if visiting "Kivrei Tzadikim" is defined as a mitzvah or not.
There was once a community in Europe who wanted to hire a particular Rabbi and in order to attract him, they told him that the Shach, the Taz and the Semah (three famous Rabbis) were buried in their town. After he became the Rabbi there, he said to them: What you told me is not true. The Shach is buried in this city, the Taz in that city and the Semah somewhere else. They answered: “What we said is true. They are not buried there, because the people there learn their words, and these Torah scholars are therefore still alive. But in our city – to our great distress – we do not learn their words. As a result, they are buried here and we therefore need you as our Rabbi.” We learn from here that it is more important to learn the words of Torah scholars than visit their graves. Visiting "Kivrei Tzadikim" does have value, as we know Calev ben Yefuneh left the spies to daven at Ma'arat Ha-Machpelah (Sotah 34b and Rashi on Bamidbar 13:22), but learning the Torah scholars works which spiritually elevate a person is certainly more important.
Furthermore, if a person lives close to the cemetery that is one thing but to fly great distances costs thousands of Shekels. Giving Tzedakah is a much clearer mitzvah than visiting "Kivrei Tzadikim." If one is looking for "segulot" (spiritual remedies), the greatest "segulah" is giving tzedakah. "Tzedakah saves one from death" (Mishlei 10:2, 11:4). This is true not only for the one who receives but also for the one who gives. The Gemara in Shabbat (156b) says that astrologers told Rabbi Akiva that the day his daughter gets married, she will be bitten by a snake and die. He was obviously very worried. On the day of her Chupah, she took a decorative pin out of her hair and inserted it in to the wall, and it struck a snake in its eye. In the morning, she when she removed it, the dead snake came out after it. Rabbi Akiva asked her: what good thing did you do? She said: On the day of the wedding, a poor person came to the door and because everyone else was busy with the meal, no one heard him. I took the portion you gave me, and gave it to him. Rabbi Akiva said: "Tzedakah saves one from death." Therefore, instead of traveling great distances to visit "Kivrei Tzadikim," one should use the money to help the poor.
Ha-Rav Shlomo Zalman Auerbach would travel from his house to Yeshivat Kol Torah in the Jerusalem neighborhood of Bayit Ve-Gan, and he would pass the national cemetery on Mt. Herzl. He would pray there, saying: "These are the graves of the righteous who died sanctifying Hashem's Name. Why should I travel far distances?"
Visiting "Kivrei Tzadikim" certainly has value but one must always weigh it against other considerations.