I agree the broad outline of the conquests is pretty accurate, but a lot of the pious Islamic narratives about them are likely fabricated or at least significantly embellished.
As for Jerusalem the accounts of its capture are, as one might expect of this holiest of cities, charged with religious overtones, but they are annoyingly bereft of concrete detail. A late Christian source makes a brief reference to a two-year siege, but gives no details or clarification. Our only contemporary source reports that “the Lord’s Cross and all the vessels of the churches of God” were taken away by sea to Constantinople for safekeeping, but of military matters simply says that having requested an oath from the Arabs that they would respect life and property, the people of Jerusalem submitted to the Arabs.
Hoyland, Robert G.. In God's Path
These are basically all of the earliest sources for Islam (other than the Quran, there are no Muslim sources for this era at all, they all appear much later).
Seeing Islam as Others saw it by Robert G Hoyland
This is why scholars are sceptical about accuracy of the theological tradition that you have been using - there are no sources for 200 years to support any of it.