So is this born-in the purple business actually a misconception?
Yes. There is, as far as I know, not one instance in Byzantine history in which a younger son who was "born in the purple" was chosen, on that basis, to succeed an older brother who was not.
That's not to say younger sons
never succeeded over older ones. Manuel Komnenos, who was born during his father's reign, was chosen to succeed him over his older brother Isaac, who was born before his father's accession to sole rule, although technically their father John II was crowned as co-emperor with Alexios long before he actually succeeded. There's no indication, however, that his birth during his father's sole reign was the reason for Manuel's succession; John seems to have simply preferred him to Isaac as a successor. The closest incident I know of to a claim of succession on the basis of "purple" birth is actually among daughters and arguably pre-Byzantine - it's in the 5th century, when Flavius Marcianus rebelled against Zeno based in part on the claim that his wife, Leontia, although the younger sister of Zeno's wife Ariadne (both daughters of Emperor Leo I), was nevertheless of superior status because she was born while Leo was emperor. (Marcianus failed.)
Being purple-born was certainly prestigious, and could confer a sense of legitimacy. The first imperial son I know of whose "purple birth" was a big deal was Constantine VII, but that's because he was conceived by the emperor's mistress who was only subsequently (and un-canonically) wed to the emperor - being born in the Purple Room was intended to be a clear statement that "this is indeed the legitimate son of an emperor no matter what the Church says." In the middle Byzantine period one tends to come across purple-ness as being more important for
daughters of the emperor, who were apparently considered more prestigious and desirable as diplomatic brides if they were purple-born. Otto the Great specifically sought out a purple-born wife for his son for that reason, although he failed to get one, receiving only a niece of John Tzimiskes instead. As far as I know the first "barbarian" ruler to obtain one was Vladimir of Kiev, and his marriage to Anna Porphyrogenita was a big deal. A purple-born bride was promised for Otto III (none other than Zoe, daughter of Constantine VIII), but his early death scuppered that plan.
So while being "purple-born" was a real thing and really was a source of prestige which might (as in the case of Constantine VII) shore up a questionable claim, there was no rule, formal or informal, that purple-born sons should take precedence over non-purple-born older sons. Even if there had been such a rule, its operation would have been rare, as situations in which a reigning emperor was survived by multiple sons, of which at least one was born before his accession and one after it, did not actually crop up all that often.