It is authentically established in the sound collections that, when the Prophet ﷺ reached Bayt al-Maqdis on the Night of the Night-Journey, he prayed two rakʿahs there; he neither prayed nor took the trouble to visit any other spot.
As for the reports of the Ascension, some are found in the Ṣaḥīḥ collections, others in the Sunan and Musnads; among them are sound reports, weak reports, and outright forgeries.
One fabricated narration claims that Gabriel said to the Prophet ﷺ: “This is the grave of your forefather Abraham—dismount and pray here. And this is Bethlehem, the birthplace of your brother Jesus—dismount and pray here.”
Even stranger is the report that says he was told in Madinah, “Dismount and pray here,” before his mosque had been built, although the site was then a graveyard of idolaters. After the Hijrah, the Prophet ﷺ settled there only because his camel knelt at that spot. Such tales, and anything like them, are agreed upon by the scholars to be sheer fabrication.
Bethlehem itself is merely one of the Christians’ churches; visiting it carries no virtue in Islam, whether or not it was truly the birthplace of Jesus.
As for the grave of Abraham, the Friend of God, none of the Companions nor the Righteous Successors ever came to it for prayer, supplication, or deliberate visitation.
The Muslims entered Syria more than once with ʿUmar ibn al-Khaṭṭāb, and many Companions later took up residence there; yet not a single one of them practiced anything of this sort, nor did the Muslims build a mosque on the site at all.
Only when the Christians seized these places toward the end of the fourth Islamic century—capturing the Sacred House after the Rāfiḍites had overrun Syria while ruling Egypt—did such innovations appear, for the Rāfiḍites are a forsaken people, devoid of sound reason.