Given you're referring to a society that existed for a thousand years it is worth breaking this down a bit. For starters, the early Byzantines did produce statues in the round. Constantine's colossus that survives in fragments in Rome and Justinian's column that depicted the emperor on horseback was outside Hagia Sophia until the Ottomans pulled it down. Going by illustrations from Western travellers you can easily find online, it was huge.

More likely you are asking why is there no surviving imperial bust from Justinian until John VII Palaiologos (now kept in the Vatican, and made by a Renaissance Italian sculptor during the council of Florence in 1430). For this we have to touch upon ideas of morality.

Actually depicting any individual in Byzantine art who was neither biblical nor canonised is incredibly rare not just because it is the churches which mainly survive. You could argue that it is due to post-iconoclastic tastes presenting depiction as a way of connecting with the divine and any figure made into an idol that was not religious, bordered on idolatry. Imperial figures and and their families dodged this because their office was imbued with religious fervour anyway. We can only assume that marble busts for the sake of pride would have counted as a sin on one's character. Artistic trends all stemmed from Constantinople after the sixth century (few exceptions: Capadoccia, Peloponnese) and other provinces followed suit, where they had marble it was incorporated into churches and homes.

Art became used mainly for two things. Religious depictions and Hellenistic patronage. Byzantines adored classical literature as the foundation of their 'paideia' their education. Constantinople until 1204 was a museum preserving ancient statues in Bronze and marble that were all viewable to the public. In Niketas Choniates' final chapter after the fourth crusade he recounts, among others, destruction of the statues of Hera, a mighty equestrian, and Herakles Trishesperos by Lysippos. The last of these was a piece of ancient Roman loot taken from Tarentum in 209 BC and thence to Constantinople in 325. All three were in the Forum of the Bull or the Hippodrome, centres of public life in the city. Others suggest that bronze statues were still being created. Niketas mentions a she-wolf of Romulus and Remus, a man wrestling a lion, sphinxes and a huge bronze Helen of Troy and many others. The horses were taken to Venice and survive to this day but from Choniates we get the impression of works of art in bronze, collected and perhaps still being commissioned.

The destruction in 1204 was the great calamity of Hellenistic revival. Bronze normally only survives when it is lost to the sea or earthquake and later recovered. They are made from the same material as coinage and necessity will entail destruction. It is important to remember that we see the relics of a defeated civilisation, we cannot assume because something does not survive now that it did not then.