TOB: A quick Bar Ilan search shows me that R’ Ovadia quotes the S’ridei Eish 47 times, the Tzitz Eliezer 8 times, and R’ Shternbuch in his T’shuvos (I didn’t check the Moadim Uzmanim, but I assume he’s Chareidi enough for you?) cites either the S’ridei Eish or R’ Weinberg himself 15 times.
Rav Ovadia quotes the Heichal Yitzchak 36 times and the Tzitz Eliezer quotes him twice. Both refer to him as “Hagaon R’ Yitzchak Eizik HaLeivi Hertzog.
Rav Ovadia quotes the Kol M’vaser 40 times, the Tzitz Eliezer quotes him 4 times and even has a T’shuvah to R’ Meshulam Roth.
The M’lamed L’ho’il is quoted 81 times by R’ Ovadia, 32 by the Tzitz Eliezer, and 5 times by R’ Shternbuch.
(The Az Nidb’ru isn’t on my Bar Ilan but I’ve seen him quote the S’ridei Eish too.)
Yes, the Zionist Gedolim may not be so popular or learned by the Chareidi masses. I will grant you that. But the contemporaries and G’dolei HaTorah, who know good Torah when they see it, respectfully quote even the L’umi/modern Poskim. R’ Goren was not quoted with the same respect because his P’sakim did not carry the same weight. I am not going to judge myself. I will say, however, that they have judged for me and I feel fairly confident being Toleh Atzmi Big’dolim on this.