How successful were Byzantium's religious missions?
This is an awfully difficult question to answer, because you are able to look at history from many different perspectives. Also it really depends on what time period you are studying. So, the best way to answer this question is to give a few perspectives.
If you are speaking of the time period of Justinian and Theodora, then they were successful to a certain degree. The empire was strong as they were able to gain some military grounds under the leadership of Belisarius. They were also able to last until the fifteen century until the Ottoman Turks took over Constantinople. Any city that could last a thousand years is very successful!
From the perspective of legacy, we need to realize that one of the most important books ever written took shape during the time of Justinian. The law Digest of Justinian was completed. This became the basis of law for the West.
If we consider their influence, then most of Eastern Orthodoxy can be considered. Take for instance, the Russian Orthodox church, which is an outflow from Byzantium. It still exists today and according to many scholars, it aided in the overthrow on communism in Russia.
Finally, if we consider all the Greek speakers who fled to the continent, then we can say that they influenced those who would start the Reformation by bring their Eastern learning.
Based on these four perspective, the religious mission of Byzantium cannot be underestimated.