Who holds that its OK to go on the Har Habayis?
Everyone agrees that we are not allowed to go into the area (inside the “sorag”) forbidden to tmei’ei met (that’s all of us!). Everyone also agrees that theoretically there are areas (if one immediately turns right= south, upon entering the entrance to the right of the Kotel) where all halachic opinions agree that it’s allowed. It’s just that most poskim believe that technically and practically, it’s easier to just say “don’t go up at all”, to prevent people from going where they halachically are forbidden, especially regarding such a severe prohibition. Among the poskim who permit men (!) to go to those holy areas (until the “sorag”) allowed only after going to the mikveh (for tumat keri), and taking off shoes, are Rav Y.M. Tokischinsky, Rav Goren, Rav Mordechai Eliahu, Rav Chaim David HaLevi, and Rav Dov Lior.