truthsharer,
The kosel is not a shul.
Actually, it is. After taking Yerushalayim in 1967, the Israelis created the Kosel plaza (there was a narrow alley there before) for the purpose of accommodating prayer. They rushed the construction of this plaza ahead of Shavuos. Furthermore, the government appointed R’ Yehuda Meir Getz and then R’ Shmuel Rabinowitz as the Kosel Rav – to oversee the proceedings that take place there. So the plaza was a space created for Jewish prayer, and a rabbi was appointed to oversee it; sounds like a shul to me. Whether you think making it into a shul was a bad idea or not is irrelevant at this point.
It’s a holy place where people daven
That’s the Orthodox perspective. The Reform movement’s theology denies the concepts of a personal Moshiach and a restoration of the Avodah of the Beis Hamikdash (G-d forbid), and furthermore they hold R”L that the Avodah itself was man-made, archaic, and even barbaric. It seems inconsistent, therefore, that they would view the Kosel or Har Habayis as holy ground. The Conservative movement severely limits mentioning the Avodah and a personal Moshiach in its prayer books.
It seems more likely to me that adherents of Reform Judaism view the Kosel as a place of nationalistic (Jewish or Israeli) significance – similar to the way Americans would view the National Mall.
My shul has existed for fewer than 100 years, but if a group of people attempted to force changes to its customs and practices in defiance of the rav, I would feel upset, even if the shul was also a national landmark.