There is indeed a 'John Asiacus' from this time who allegedly converted many pagans - but he is better known as John of Ephesus, as he was the bishop of Ephesus. In his Ecclesiastical History he also described himself as 'John of Asia' (an understandable choice, since Ephesus was in the Roman province of Asia), which explains why someone Latinised his name into John Asiacus. However, I haven't encountered any modern historian who described him as John of Asia - at best, I've only seen Anthony Kaldellis using 'Yuhannan of Amida' to refer to the same author, as Yuhannan is the Syriac form of 'John', a choice Kaldellis made because Syriac was John's native language, whilst Amida was his birthplace. Either way, using John of Asia is confusing and not exactly a sign of historical vigour from this polemicist.

As for the 'crusade' itself, we have to turn to John's own writings, as he is the only source for these seemingly momentous events. Unfortunately, the portion from his Ecclesiastical History covering this is actually lost, but luckily for us they were used by later Syriac historians. From the Chronicle of Zuqnin (or the Chronicle of Pseudo-Dionysius, I haven't worked out which name is the scholarly consensus yet):

The year 853 (A.D. 541/2): God’s grace visited the countries of Asia, Caria, Lydia and Phrygia by the zealousness of the victorious Justinian. It proceeded from him abundantly by the mediation of our humble self, that is John of Asia, to the effect that by the power of the Holy Spirit 70,000 souls were instructed in the faith and turned away from the error of paganism, idolatry and the celebration of the demons to the knowledge of the truth. They were converted, signed with the Cross and baptized in the name of our Lord Jesus Christ. Having renounced the error of their fathers they joined the ranks of the Christians. (For) the expenses and the baptismal robes the victorious (emperor) gave abundantly and he also took care to bestow upon every one of them a trimision. So when God opened the minds of all of them and enlightened their understanding with the truth, together with us, they destroyed their temples with their own hands and hewed down rows of their idols, cast down the sacrifices set in order everywhere, and overthrew their altars which were stinking and impure with the evil smell of blood from the sacrifices to demons, and hewed down the innumerable trees of their cult, while deriding the entire error of their fathers.

Thus the sign of the Cross of salvation was set up everywhere among them, and God’s churches were consecrated with much zealousness and care in every place, on high and habitable mountains, in valleys, and in other places, where paganism had disappeared. As many as 96 of them were built and erected, and 12 monasteries, in the places where paganism had disappeared. Where the word “Christianity” had never been pronounced since the foundation of the world until (this) time, (there) 55 churches were built (with money) from the public but the new Christians built 41 at their own (expense), whereas the victorious emperor through our humble self abundantly gave silver and linen garments, books and brass (vessels) gladly and willingly from the public treasury.

Sounds pretty impressive right? But note how the number of churches built don't add up to the figures you provided! The polemicist must then have taken the figures from an original section of John's Ecclesiastical History that does survive independently:

[John's friend Deuterius], through a period of five-and-thirty years, was fellow-labourer with John in instructing the heathen in the provinces of Asia, Caria, Phrygia, and Lydia, where together they built 99 new churches and 12 monasteries.

Of course, numbers do change during the manuscript transmission process (particularly as we only have the first account relayed through another anonymous author), so this is far from the biggest fault of the polemicist here. The problem is in how they interpreted the source to depict a crusade that was seemingly aimed against pagans. Strictly speaking the facts presented are correct (according to John himself), but its interpretation is an ahistorical one, particularly the use of the word 'crusade' to describe the conversion of pagans. Describing John as an 'inquisitor' is doubly anachronistic and speaks more for the polemicist's own views than John's actions. Beyond that, as we have no evidence that John ever received an official title nor that he had powers to violently coerce people to convert, I wonder how on earth we can arrive at the same conclusion as the polemicist.

From what we do know, I can say that John was writing in the 580s, when he had fallen from imperial favour but still retained some reservoir of goodwill towards Constantinople. He had since then been imprisoned a couple of times and had his hopes of converting the imperial establishment to his brand of Christianity (miaphysitism) dashed again and again. When composing his Ecclesiastical History, John must have looked back at his career with some sense of (bittersweet?) pride, in the process shaping, exaggerating, or otherwise manipulating the resulting narrative. Indeed, the fact that no other source mentioned this so-called 'crusade' would suggest that it lacked official support and possibly only took place on a much smaller scale. In an age when we are literally overflowing with primary sources, Emperor Justinian's enemies and allies would have surely mentioned this massive expenditure of resources - and if 70,000 people were indeed converted, then surely it would be trumpeted in imperial propaganda?

As a result, I am very much in agreement with Volker Menze that John of Ephesus' efforts, however great its personal value to the elderly bishop, were not sanctioned directly by the emperor, but probably by other miaphysite members of the aristocracy. As Justinian sought reconciliation with the miaphysites at the same time, he took advantage of the mission's success by granting patronage to the missionaries involved and the converts. This of course remains only one interpretation and I suspect that someone working specifically with Syriac texts will have rather more to say on this than me, but I hope that it is still enough to show why John of Ephesus' words must be interpreted very carefully and that regardless of how we interpret them, the polemicist's account is not historical, not helpful, and certainly not one that does justice to the complexity of the sixth century.