So this is a complicated question in part because you first have to distinguish between “people ante incorporation into the empire” and then “people post incorporation.” Then which phase of “empire” you are talking about, which I’m going to divide into 3 periods: the Roman empire of the Republic (3rd c. BCE - 31 BCE), the Roman Empire (31 BCE - 5th century CE) and Romania (I.e. the Byzantine Empire 5th - 15th century). These aren’t perfect divisions, entirely distinct periods, or the only divisions I could use but they work for my purposes.

So first things first, did unconquered people want to be part of the Roman Empire? I’d say in general, no (duh?). Rome conquered their empire largely through military conquest (though they were bequeathed Pontus in the dead king’s will which is just one of those historical facts that consistently amuses me). However, particularly in the east, the lands they conquered were spear won kingdoms often ruled by ethnic outsiders so the native resistance was minimal as it was exchanging one person you paid taxes to for another (I can not overestimate how small Rome’s governing footprint was, especially early on).

(Jewish resistance first under the Seleucids in the 2nd century BCE before Roman conquest and later against the Romans in the first and second century CE is a useful exception to this rule due to the unique characteristics - a lot is because of their weird monotheistic god - of the Jewish people that made them hard to incorporate).

Okay now let’s look temporally. So first: most of the Roman Empire was conquered under the Republic. Also, Roman rule of their lands at this time was pretty shitty, as in they didn’t do much. During this period, I highly doubt the conquered peoples felt part of Rome in a material sense as in Roman rule wasn’t very visible and they definitely did not on an identity level (which will be important shortly).

During the Empire, things start to change because Roman government had changed. The Empire put a greater effort in incorporating the people. Augustus’s face was ubiquitous: cults were built to him, statues put up in the forums/agoras of Roman cities, his face was on the coins. His government was present. You could address your grievances with his magistrates. Maybe they built your town an aqueduct. The sons of your town would join the army. The army garrison nearby meant that the city was mostly secure.

Maybe you missed being independent but you also have to admit this has its perks.

Initially maybe you, random guy in Syria or wherever, are only putting up with this, but your son buys into it more as it is all he knows. And then his son joins the army and gets citizenship. And his son identifies as a “Roman” and legally is one even though your family lives in Syria or some other backwater and nobody in your family has ever been to the Urbs Aeterna. It is this process of good, regular governance along with the building of common structures to give Roman cities a common look, and consistent reinforcement of Roman rule as good and valid via propaganda, and just the habituation of life under this system that is the process that we shorthand as Romanization.

These people not only wanted to be in the Roman Empire, they were Roman. See Ando, “Imperial Ideology and Provincial Loyalty” where this whole process is laid out in incredible detail. Particularly the first chapter where he lays out all the theory for how this process works.

Where does Ando see this end up? Well the opening words of his book sum it up quite well:

“No date identifies that moment when Rome ceased to rule her subjects through coercion and began to rely on their good will; no event marked the transformation of her empire from an aggregate of ethnic groups into a communis patria.”

What are the implications of the “communis patria” and therefore for this third temporal period that I have divided Rome imperial rule into? For that I will turn to Kaldellis and his monograph, Romanland. His book is about the Byzantine empire which we both acknowledge is the continuation of the Roman Empire for another 1000 years yet are simultaneously denying by instead giving it this stupid name. A name, might I remind you, The Empire never used. What did they call themselves? Romania (not pronounced like the modern country - you should put the stress on the I). Or Romanland, inhabited by the Romanoi or Romans.

Basically Kaldellis’s argument is that we ought to understand Romanland as the Romans did (emic, not etic) and the evidence is overwhelming: this was no longer truly a multiethnic empire, it was a nationstate filled with ethnic Romans. Why isn’t it an empire? Because as a political system, an empire is built on the domination by a dominant ethnic group of a bunch of ruled peoples.

Christianity was definitely a binding agent as was the person of the emperor and, as is the case in many nations, they were likewise united by what they were not. (e.g. after the 7th and especially 8th century, Romanoi were not Muslim)

And so for these Romans: yes they still wanted to be part of the Roman Empire (though I suppose if we accept every part of Kaldellis’s argument at this point it was no longer truly an empire).

Apologies for any typos or grammatical mistakes.

Other Sources and suggested further reading beyond Kaldelis or Ando on this same issue, or that approaches it or some element of it quite differently:

If you are interested in slapdash functioning of Roman imperial rule under the Republic, I found Tan, “Power and Public Finance” to be very informative.

For why Rome created an empire, Harris, “War and Imperialism” remains a classic (although Harris is a creep)

Spawforth, “Greece and the Augustan Cultural Revolution” is a different kind of look at Rome’s “romanization” of the provinces. Though aspects would be particular to Greece and Rome’s own relationship with Hellenic culture the book argues for a clearly coercive framing for Rome’s control and an imposition of a particular cultural expression

Finally, I like Wallace-Hadrill’s “Rome’s Cultural Revolution” and how it gives a very nuanced look at how this cultural transformation would have actually played out on the ground as it never is solely a situation of metropole transforms province. It is always a give and take as they transform each other.