Basically, he did both, though more the former than the latter. Our early sources on Constantinople are not the best (and what we do have is more church-oriented), so a lot of it is based on conjecture, but we can vaguely understand how the process worked.

As you rightly say, the idea of the imperial court and bureaucracy creating "new Romes" in the frontier regions was not new, even including the construction of certain buildings imitating Rome. Licinius, whom Constantine defeated in the civil wars, had even done this in Byzantium (probably one of the motivating factors for Constantine's move there). Of course, these structures are always shifting over time, but they remained relatively fixed after Diocletian's reforms, despite their varying locations. With that said, Constantine certainly reformed the administration and centralized it more directly under imperial authority. It's hard to say too much, given the sources, but it seems like he mainly expanded and modified the existing bureaucracy rather than adding new institutions de novo. And this was also largely separate from the founding of Constantinople itself, with the bureaucratic center being instead tied to wherever the emperor was, as it had been under the tetrarchs and before.

But what Constantine did was quite different, signaled clearly by his creation of a new senate. According to Hesychios (fragments of whose patria, a sort of mythologically infused history of the city, survive), they were partly made up of imported senators from Rome, for whom Constantine built the kind of luxurious manses they would expect back in Italy. But they were also drawn, as you suspect, from the provincial elites of the Aegean world. The sources are often very degrading toward them (describing them as sons of butchers, cheesemongers, etc.), but they seem to have been perfectly reputable local elites who were now promoted to senators by Constantine and given a cushy place in the new capital. Now, the senate did not have much authority with regard to the military or foreign policy, but they were certainly important to the local civilian administration, playing much the same role they had settled into in Rome.

Sources/Further Reading

Raymond Van Dam is probably the most prominent scholar who covers this topic, and both his Rome and Constantinople: Rewriting Roman History during Late Antiquity and The Roman Revolution of Constantine are great places for further reading, as well as sources for this answer. I'd also direct attention to the Cambridge Companion to the Age of Constantine, especially the article on "Bureaucracy and Government" by Christopher Kelly.