I rather disagree with this assessment; for starters, the 'Schola Palantina' was a unit - a 'regiment' if you will that had a particular name, along with several other 'Tagmata' regiments to have the same - its not a type of soldier at all, though they were, to the best of my knowledge comrpised of mostly if not exculsivly heavy cavalry. a 'Vexelatio' isn't a unit either, persay, but again the name for squadron of cavalry - but with no clarification as to just what sort of cavalry it is.
Secondly, the date of Julian the Apostates' reign began in 361 (and ended the next year) - but the first distinct use of heavy cavalry begin in the time of Trajan, the word 'Cataphract' frist comes into use under Hadrian - and there are even earlier murmuring as Josephus mentions kontos (heavy lancers, rather then the normal javelin using cavalry) in the Judean war of Vespasian, and certainly by the crisis of the third century heavy cavalry have become important, not only militarily, but in terms of prestige - as early as the 250's you have Gallienus establishing units of cavalry corps based at major cross road outposts in order to serve as mobile reserves for the army (and here you can see the very beginning of the border guard/field army system of Diocletian and Constantine in some respects) and in the wake of Gallienus you have have the usurper Aureolus attempting to sieze power not from a position of provincial governor or Legionary army commander, but as a man whos at the head of these cavalry reserves - and indeed the next two full emperors, Claudius and Aurelian are both cavalry commanders themselves as well, the latter being recoded as using Clibinarii (a very heavy type of cavalry indeed) against the Palmyran rebels, though the winning cavalry tactic in those wars was with the use of light horsemen.
However, trying to relegate Roman heavy cavalry to Julian and beyond is a mistake, just as much as trying to make them Barbarians.
Were barbarians used? Sure they were - though as a specific example, the Isaurians you mention weren't barbarians at all, but rather a Hellenized people from south Anatolia who had a reputation that we well might associate with the term 'Red neck' today; hardly barbarians, even if not quite the most cosmopolitan inhabitants of the Empire.
The use of barbarian foederati is a complex one however, and its still not entirely understood in detail, but its safe to assume that even if very large formations and elements of the armies came from barbarians, where was a similar proportion of troops who originated from the Romanized or Hellenized provinces of the Empire, and that these soldiers would have a presence in all levels and unit types present, out side of those exclusively federated and based around a non Roman ethnicity. The time of Justinian is particularly interesting, as it is about this time that Barbarian mercenary use was at its height - having been the cause of the fall of the Western Empire, but in the east you see Generals procuring barbarian forces use on campaign, but the retention of cavalry corps in the capital manned not only by native Romans - but by the local aristocracy of Constantinople itself.
To summarize it shortly - the Romans and Byzantines were major users of all potential forces at their disposal, even if the leadership then provided was often dismal, these forces included all manner of soldier, both citizen and barbarian, and the forms either of these took could, and did manifest itself in just about any military roll you might think of for either.