Ray Van Dam's book Rome and Constantinople: Rewriting Roman History during Late Antiquity does a pretty good job of answering this, specifically his first chapter on Rome, and I'll be using it below.

That said, you're assumptions are absolutely correct and you're questions are hitting at exactly the right spots. Rome was incredibly unsustainable and, economically speaking, a supply sink. The city was estimated to have around 1 million residents at its peak (the early empire), bigger than a lot of industrialized cities even in the modern world. And of course, these were not people who made their own food, and so needed supplies from the rural regions of the empire. The vast majority of grain shipments to Rome came from Africa, notably Egypt. Van Dam estimates that at its peak, Rome was importing at least 220,000 tons of grain each year, and this is admitted to be a very low estimate. And this isn't accounting the amounts of olive oil and wine, two staples for the ancient diet, which were being brought in. As an interesting fact, at the end of the 2nd century, emperors began to add olive oil to the grain dole, and in the third century, pork and possibly wine were given. Additionally, a free market system was not moving these supplies, but rather the imperial administration was deliberately managing Mediterranean shipping specifically to sustain Rome. With all of this in mind, consider Van Dam's statement that a free market system could only sustain a population center of 10,000-20,000 people in the ancient world.

In fact, the capital's population was so comically high, that it couldn't come close to sustain itself off its own birthrate, but was overwhelmingly reliant on immigration from rural regions. This was particularly due to the comparatively higher mortality rates of urban regions prior to industrialization, which entailed things like increased disease and lack of sanitation. One estimate cited by Van Dam provides a figure of 10,000 "or more" people being imported into the capital each year.

The worst part about all this, at least to a provincial, was that Rome was not a manufacturing center or a producer of anything productive to the rest of the empire. Apparently, the only major commodity that Italy mass exported at the peak of Rome was sand, to ballast ships.

The obvious question to all of this is why? Vam Dam argues that the reasons for keeping Rome at such a massive population, and even attempting to enlarge it, were entirely ideological. As he puts it:

In terms of the effort required to sustain the scale and complexity of its logistical supply for so many centuries, the size of Rome made little sense in this sort of underdeveloped, unmechanized, preindustrial agrarian economy. But in terms of articulating and projecting a particular grand narrative about Roman imperial history, it was vital. The enormous size of Rome confirmed a particular myth about emperorship, it provided a venue at which emperors could perform their emperorship, and it was a visible guarantee of security and peace for outlying provinces. Great size might be a powerful symbol. The size of Rome hence represented an ideological commitment to a particular historical narrative, a particular notion of emperorship, and a particular idea of empire. (p.18)

Essentially, Rome was a physical manifestation of imperial power.

Firstly, it provided a centralized location for the emperor to interact with and ensure the cooperation of the wealthy senatorial elite of the empire. These senators were likewise often sent out into the provinces to govern, and the emperor's continued friendship with them was used to show adherence to republican tradition.

Rome also served as an audience to imperial displays of power. Public appearances (at the colosseum for example) or the sponsoring of monumental architecture was best done in the capital, with a densely populated audience in the hundreds of thousands.

The city likewise sent a message to the provinces: that in return for the exporting of their production, they received back from the capital peace and security. Of course, a more cynical interpretation of this is that the transfer of wealth to Rome represented a reminder of the provinces' subordination to the capital.

These ideological motivations become all the more apparent in the 3rd and 4th centuries. Due to the military instability of the 3rd century, emphasis was placed more on the army rather than the senate for imperial legitimacy. This led to many emperors being proclaimed by soldiers on the frontiers rather than in Rome. Constantine the Great, for example, visited Rome only 3 times in his 30 year rule. The growth of the army also meant the deprivation of migrants from Rome. Its impact was especially strong because the army only recruited young men at a rate similar to Rome's peak levels of immigration.

And there was also the transferal of the capital from Rome to other places, namely Ravenna, the tetrarchic capitals (Trier, Milan, Nicomedia, and Sirmium), and Constantinople. In Constantinople specifically, we find deliberate attempts at sustaining this new capital similarly to how Rome was kept alive. Though, this was done entirely at Rome's expense. After all, the imperial prioritization of Constantinople entailed the diversion of African grain away from Rome. In fact, Constaninople at its peak imported more grain from Egypt than Rome ever had.

All this compounded to Van Dam's assessment that "In just over two centuries Rome had lost almost 95 percent of its ppopulation from its peak of one million residents." As he states, in the 5th century, it had a population of 500,000, at the end of that century, it was at 350,000, and by the early 6th, it had been around 60,000.

The ideological approach that Van Dam takes is just one perspective, but it certainly does explain a lot about the nature of the capital and its strain on the empire. But to sum it up, yes, Rome required a massive amount of resources from the rest of the empire for little material return. However, the city was vital to the emperors in order to show off their authority and reinforce their legitimacy in a centralized location.