That was my thought precisely ! Rousseau's Social Contract could definitely take a different dimension if he can draw from the Inca model and contrast it with the European one.
Also, I was wondering about immigration. If the Inca Empire manages to remain stable and prosperous while retaining decent equality among its population, then wouldn't that make it attractive for XIXth century migrant populations? Especially the ones crossing through the Pacific....
That seems possible, but immigration to the Inca is going to be strict and on the Inca's terms in all likelihood. It's also questionable just how popular immigration would be for the average poor European or South/East Asian when you consider how authoritarian the Inca Empire was towards its populace. Sure, on the one hand you'll never starve. On the other hand, your entire community could be upended at the Sapa Inca's whims, military service is mandatory, and you'll never own much of anything beyond the clothes on your back and maybe your family home. Immigrants would have to reconcile their understanding of personal wealth and communal responsibility with everything they know, and the ones that are most familiar with these ideologies(serfs, peasants, slaves) are those least likely to be able to migrate to the Inca Empire. On the other hand, the Inca might actively court immigrants considering their demographic issues which even two hundred years of complete peace are unlikely to completely resolve unless they manage to cheat code their way into vaccination and germ theory waaay ahead of anybody else. While I think that they'd be strongly motivated to make these kinds of discoveries(considering the amount of political pressure there'd be behind investigating the diseases that ravage them on and off) it'd be incredibly unlikely to occur unless a foreign state or individual introduces vaccination to them. The diseases they're trying to control are so deadly to them that being able to research them would likely be an impossibility unless you've got a survivor(low odds) plus the resources and conditions behind them(low odds) to facilitate said curious investigate soul(low odds) to be a radical innovator. As much as I cape for the Inca that's incredibly tall odds. Which isn't to say impossible buuuuut...
Back on the topic of immigration, the Inca are gonna be conflicted on it unless their society's been revolutionized via Old World contact. On the one hand, the Inca have a serious demographic problem barring, I dunno, an early Inca embassy to the Chinese court or to a major Indian state that sees them share knowledge of inoculation with them. Considering inoculation reached Western Europe in the early 18th century, any scenario other than early Eastern contact means a demographically troubled Inca state. Moving on from inoculation(for real this time), the Inca need people. But these people are going to be foreigners. Foreigners that are utterly unused to Inca institutions and society, which makes the attractiveness of the Inca questionable. But the Inca have likely had some amount of experience with foreign migrants from Europe that acted as advisers, artisans, and traders, IMO it'd form the basis of the Inca's approach to them. IE, the short leash, of questionable attraction to most immigrants. Between the societal differences, the religious differences, and the torn attitude of the Sapa Inca to foreign elements that threaten the coherency of his realm, I think that'd immigration would be largely limited from Europe. The British are too used to constitutionalism, the French rarely migrated IOTL, the Spanish have their own colonies, ditto for Portugal, the Dutch are largely traders, etc. There's simply too many alternatives and too many reasons to say no to the Inca for Europeans to find settlement there attractive barring romanticists and radicals that overly idolize the Inca for whatever reason(probably their social safety nets if anything at all) and that number is likely to be incredibly small. East Asians I find more likely if only because Chinese and Inca society would find more in common than the Inca would with Europe and the lesser emphasis on religion. I can imagine destitute Chinese in major port cities of China biting at a chance to migrate to the Inca Empire with offers of land and work and familial prosperity in a 19th century as awful as OTL's for China. Outside of China I don't know much of anything about the rest of East and South Asia beyond the superficial to offer commentary or speculation on, but I'm dubious of large scale Hindu migration due to the religious aversion to being separated from the Ganges that I see commonly cited.
IMO the largest source of immigrants would be Africans, though not in the 19th century, but during the peak of the slave trade. I'm gonna go on a bit of a reach here, but if the Inca have bullion aplenty and have a severe demographic problem whereas other European states have a great desire for said bullion and are already engaged in the slave trade for plantations, what's the possibility that the Inca end up being a major consumer of slaves? There's the possibility that Africans are horrifically exploited akin to Europe's plantation slavery by the Inca and follow Europe's example, but that would be converse to the Inca's modus operendi of integrating the conquered into the Inca state so as to maintain the stability of the realm. Considering the strain the Inca would already be under demographically, the Sapa Inca would quickly become Not Amused a few slave rebellions later and ditch their experimentation with slavery. I think that in a scenario where the Inca end up participating in the slave trade, the Inca would see Africans as a means to repopulate the lands with new Quechua rather than as exploitable labor. This'd mean purchasing slaves from European traders, transporting them to Ayllus, and settling individual families throughout. A few generations later and you've got a sizable minority of Afro-Quechua that have blended into their local communities akin to any other natives that have been conquered by the Sapa Inca as well as ample intermarrying. Between emancipation, societal pressure, and relative improvement in quality of life, I think such experiments would prove successful and see the Sapa Inca allocate more resources into these projects. There'd be more frequent outbreaks in this scenario versus one where the Sapa Inca commit to isolation, but the Empire would be less and less impacted over time. Amusingly, you might see the Inca correlate immunity to African blood and see mixed Afro-Quechua Emperors a few generations later, or even an early variant on Paraguay's intermarriage policies that sees the majority of the population consisting of mixed African/Quechua over the course of a few centuries between the importation of slaves and the demographic troubles the Inca face with its native populace. The Inca are in a position to enforce these kinds of things, as unlikely as it'd be.