According to my understanding no one is saying Josephus embellished the details.What would Josephus have to gain by writings the way he wrote or embellishing the details? He was a Jewish man. How is the Christian perspective on what he said interpolation?
If you read the article linked to in full it should answer most questions you have, for example take this part;
'Josephus wrote all of his surviving works after his establishment in Rome (c. CE 71) under the patronage of the Flavian Emperor Vespasian. As is common with ancient texts, however, there are no extant (surviving) manuscripts of Josephus' works that can be dated before the 11th century, and the oldest of these were copied by Christian monks.[26]'
In other words we don't have Josephus' original copy, we have a copied version, most probably edited by Christian monks who had plenty to gain by editing it to suit the Christian narrative.