from what i understand the status-quo is that only Muslims can pray there and any Jew visiting the site cannot utter any prayer he cannot bow and i'm sure there are other things that are forbidden.
so the question is why is this the law in Israel? why israeli politicians are not planning to change it?
there are 2 possible reasons to avoid changing the law
1) the charedi parties that have been part of most israeli governments prefer to keep it in Muslim hands so that no Jew will enter places that are forbidden for Jews. There is a halachic disagreement if it is permissible to enter Har HaBait. The religious zionists rabbis claim it is permissible except of few places where the Kodesh Kodashim was. The charedi rabbis forbid Jews to even enter the area. This is very rough description of the problem and i appologize for simplifications.
here are some links with more in depth description:
http://5tjt.com/going-onto-har-habayit/
https://www.ou.org/torah/machshava/tzarich-iyun/tzarich_iyun_har_habayit/
http://www.hakirah.org/Vol%2010%20Messner.pdf
israeli Rabbinate approach to Har HaBait: http://www.jcpa.org/jpsr/s99-yc.htm
the possibility #2) is that we are afraid of what the muslims will do (pretty much what they are doing now).
I would suggest to ban all prayers on Har HaBait - not to discriminate. I understand that this is not a practical thing to do but i think wee should be thinking about all possible solutions.
just a thought,
