Answer
Shalom Rav! Because of the holiness of Har HaBayit, we have a prohibition to allow any desecration there, irrelevant of whether we go up or not. If we aren’t allowed because of impurity, how much more so are the gentiles, who are even more tamei, forbidden there, and we must try and prevent all desecration. Accordingly, it would definitely be better that Har HaBayit remain empty rather than have gentiles go there. In fact, if we wouldn’t have the political problem of the mosque, we could and must revive the avoda of the korbanot, at least korban Pesach, for even if we all have the chazaka of tum’at meit (most have been within 2 meters of a grave or in a hospital where there is a corpse), the rule is that if most of Israel is impure, than that issue is suspended, and we can revive that sacrificial service even being tamei (see Rambam, hil. Korban Pesach 7, 1-2).
Technically and strategically, I think we should start at least by demanding that the Arabs not play soccer there, for the international community (aside from the illogical anti-Semites!) will definitely see the hypocrisy of such desecration (if they say the entire Temple Mount is holy, so treat it as such- if it’s not so holy to them, than they must at least relinquish and give in to us at least that portion on which they play soccer!).
In general, it must also be publicized that there are areas (if one immediately turns right= south, upon entering the entrance to the right, above the Kotel) where all halachic opinions agree that it’s not prohibited. It’s just that many rabbanim believe that technically, it’s easier to just say “don’t go up at all”, to prevent people from accidentally going where they halachically are forbidden.
