In the Revelation Study, Pastor Armstrong mentions that the average person living in the Millennial Kingdom may never see Jesus in his Temple because only priests can go into the "Holy of Holies," but doesn't Peter call all believers priests?
All believers in the church today are called priests, as you said, because our bodies collectively are the temple of God (1Cor 6:19). Christ dwells among men on earth by His Spirit in the bodies of believers.
In the Kingdom to come on earth, however, the temple will no longer be our physical bodies. Instead, a new physical temple building will be constructed for the Kingdom period, according to Ezekiel and Isaiah, and the Lord will inhabit that building as He once did in Israel’s past.
The best description of Jesus entering the Kingdom temple is given in Ezekiel 43. The prophet was shown a vision of the future temple (which Ezekiel calls the “house” of the Lord) as the Lord enters to reign:
Notice that in the Kingdom, Jesus will appear as the Shechinah glory of God, not as a man as in the days He first walked the earth. Furthermore, the Lord declares He will never leave His throne in the house of the Lord again. Therefore, in order for Jesus to be seen during the Kingdom, a person must have access to the Holy of Holies or some other part of the Millennial Temple. You may find the Ezekiel Study helpful.
Furthermore, during the time of the Kingdom, the Lord institutes a new Kingdom Law. The Kingdom law brings with it a new priesthood (see Heb 7:12). The requirements of the Kingdom law are given in Ezekiel 44-46, and in Ezekiel 40-43 we learn that Levitical priests will once again live around and officiate in the Kingdom temple.
Isaiah tells us that all the Kingdom nations will have the opportunity to witness the glory of Christ in the temple:
Notice Isaiah says that all nations will see Him in the temple, and the Lord will take some of the people of these nations as His priests. Therefore, priests will include Gentiles in this period. If we are selected to be a priest in the temple, then we will have access to the glory of Christ in a regular way. Otherwise, our access will be more limited as a worshipper.
Therefore, we know the nations will see Christ’s glory in some limited fashion, but what they experience will not be what the first-century disciple experienced. The world will see the glory of Christ in ways similar to how Israel saw the Shechinah glory of God in the Old Testament. Periodically and in a limited fashion.
In summary, the Church saints are priests today, because the temple of God resides in our bodies, but in the Kingdom, the temple changes as does the law, and therefore the priesthood changes as well.