I collected some reviews of Norwich's work for a previous question a few months ago, so I can post them again here. Academic historians seem to agree that Norwich is an excellent writer who tells an interesting, lively, and even gripping story, and that his books are a great introduction for general readers as well as history students. However, Norwich himself, in his books about Byzantium and about all of the subjects he writes about, always points out that he isn't really an historian, he's just an amateur with a lot of time on his hands and all he wants to do is tell a good story.

For academic purposes, the consensus among historians is that Norwich can be a bit too credulous and uncritical of his sources. He knows the primary sources well (but admits that his Greek is not strong, so he has to read translations), but he mostly limits himself to the well-known chronicles and ignores other kinds of sources (whether literary or archaeological). He's also not up-to-date with recent work done by academic historians, so the secondary sources he uses are often obsolete by many decades.

It's actually a bit difficult to find scholarly opinions about him, since academic writers tend to ignore him more than anything. But here are some typical reviews:

“Although Norwich provides a lively narrative readers may enjoy, students seeking a reliable introduction to Byzantine history will need to look elsewhere.”

Alan Harvey, review of Byzantium: The Apogee, in The Historian 55, no. 3 (1993), pg. 555

“John Julius Norwich's Byzantium (3 vols.: London, 1988-95) is an uncritical compilation of largely obsolete work (which, however, 'makes no claim to academic scholarship,' I, 28)”

Warren Treadgold, A History of the Byzantine State and Society, Stanford University Press, 1997), introduction, pg. xv

"Norwich's narrative is so lively, certainly partly because of his own skill as a writer but also very largely because he relies so much on lively Byzantine stories; and his history is so bad because he is unaware of how those stories operate - he believes them. So Norwich's narrative is, in its way, very Byzantine: it is very readable and not to be trusted."

Roger Stone, "Byzantine Chronicles", in The Medieval Chronicle 6 (2009), pg. 34

So, his "A Short History of Byzantium" and his other works about the Byzantines, and also his works about Venice and Sicily (and presumably his other writings, but especially the ones on those three subjects) are good introductions and "broadly correct" as you say, correct enough for beginners looking for a good read. But if you were more than a casual reader, and you wanted study Byzantium in depth, Norwich would not be very useful.