The primary issue with your question is that you are assuming that the Dome of the Rock itself wasn't used as a place of worship. It was, although its purpose always was to be more than just a "friday mosque."

/u/Farquier is right to note that there was already a site for Muslim worship in Jerusalem before the construction of the Dome of the Rock, but the details on it are quite hazy.

The Dome of the Rock is constructed at an incredibly interesting cross-roads for the early Islamic community - namely, during the time of second fitna, or civil war, which proved to be a major formative event for the young community and from which a number of major reforms developed. The Dome of the Rock itself bears an inscription dating it to the year 72 AH/691-692 CE, but we don't actually know if this was the year the building's construction was started, or when it finished.

It seems much more likely to be the latter, however, and it seems like it may well have been constructed by 'Abd al-Malik to be a new pilgrimage site more than anything else, as his rival - 'Abd Allah b. al-Zubayr - controlled the holy cities of Mecca and Medina during the second civil war. In this way, the construction of the Dome may have served a great number of legitimizing purposes for 'Abd al-Malik, not least of which was his attempt to demonstrate to the rest of the Islamic world that he was actually the rightful caliph, not Ibn al-Zubayr, and that he controlled a pilgrimage location more important than those in western Arabia.

These issues surrounding the building's construction are discussed at length by Sheila Blair (who argues for the date as a beginning date, one which I definitely disagree with) in “What is the Date of the Dome of the Rock?,” in Bayt al-Maqdis: ‘Abd al-Malik’s Jerusalem, Vol. 1, ed. Julian Raby and Jeremy Johns (Oxford: Oxford University Press, 1992), and Jeremy Johns, “Archaeology and the History of Islam: The First Seventy Years,” Journal of the Economic and Social History of the Orient, Vol. 46/4 (2003).

So part of the reason for this was that there already was a mosque on the Temple Mount, probably built by Umar or Muwaiyah I. This early mosque was replaced by the current Al-Aqsa mosque, depending on who you ask either in the reign of Abd al-Malik concurrently with the building of the Dome of the Rock or in the reign of his son al-Walid. You don't hear as much about the Al-Aqsa mosque in textbooks mainly because it was heavily rebuilt over the years and badly damaged by fire at one point, making the Dome of the Rock a better example of Umayyad architecture, and because it is more common to discuss the Great Mosque of Damascus since that's a much better-preserved building from that period. Also note that strictly speaking any enclosure or building with a quibla would be considered suitable for prayers. In any case the building program of the Temple Mount under Abd-al-Malik and its motivations is an interesting historical issue in its own right.