First, Temple-mount is not equivalent to Masjid-al-Aqsa, the latter only happens to be located on the former. A mosque in general and Sharia in particular is therefore not a factor in the calculation of restrictions.
Secondly, the question in the title is from the Muslims point of view, whereas the contents of the question are about Jewish - rather Israeli - policies.
It is narrated that a delegation of Christians from Najran visited the Prophet (puh) and prayed in the mosque facing eastward. The Prophet (puh) witnessed them praying and ordered his companions to let them pray. This sets the precedent for people of the book to pray in the holy mosque of Medina. This incident is reported by the historian Ibn Ishaq, and included in Tafsir-ibn-Katheer, in exegesis of 3:61.
قال ابن إسحاق: وحدثني محمد بن جعفر بن الزبير، قال: قدموا على رسول الله صلى الله عليه وسلم المدينة، فدخلوا عليه مسجده حين صلى العصر، عليهم ثياب الحبرات؛ جبب وأردية في جمال رجال بني الحارث بن كعب، قال: يقول من رآهم من أصحاب النبي صلى الله عليه وسلم ما رأينا بعدهم وفداً مثلهم، وقد حانت صلاتهم، فقاموا في مسجد رسول الله صلى الله عليه وسلم يصلون، فقال رسول الله صلى الله عليه وسلم " دعوهم " ، فصلوا إلى المشرق
Ibn Ishaq said, "Muhammad bin Jafar bin Az-Zubayr said that, The (Najran) delegation came to the Messenger of Allah in Al-Madinah, entered his Masjid wearing robes and garments, after the Prophet had prayed the 'Asr prayer. They accompanied a caravan of camels led by Bani Al-Harith bin Ka'b. The Companions of the Messenger of Allah who saw them said that they never saw a delegation like them after that. When their worship time came, they stood up to perform their worship in the Prophet’s Masjid. Messenger of Allah said: “Let them (worship)” and they prayed towards east.
Consequently, the juridical discussion by jurists have revolved mostly around the status of polytheists rather. Notably, Abu Hanifa, even allowed the polytheists to enter the holy Meccan mosque.
The Islamic law permits the Jews and as for politics, Muslims in Palestine are in no position to determine, let alone enforce visitation policies for the Jews.
Jews are internally divided on legality of praying inside a mosque. The most influential Rabbi Maimonides allowed praying inside the mosques, possibly himself prayed in a mosque. It is therefore hardly a matter of Shariah or Halacha. There are three main motivations for their self-imposition of restrictions to praying on temple-mount.
That the Jews are impure (corpse impurity) and therefore prohibited by Halacha to enter the holy Temple mount. Purification can only be accomplished by sprinkling water with ashes of a red heifer.
That because the exact location of the temple on mount is unknown, someone may inadvertently step inside the holy place.
The risk of situation going out of hand in case of a head on confrontation inside an already charged environment and the consequent reaction of over a billion and a half Muslims around the world.
Personally, I think neither one of these factors alone is the basis of restrictions; they all work in tandem. The risk of blow-back is there but since its not worthwhile anyway because of the religious restrictions (its not time yet!) theres no need to experiment. On the other hand, not all Jews respect Halachic restrictions, some actually demand for the right to pray and in that case the security and political risk calculations of the Israeli government are holding back the horses.