There is a long section on Wikipedia which posits an early date (c. 1910) for the emergence of HIV under the brutal conditions of the Belgian Congo. Specifically: sudden urbanization, the sex trade, and unsafe mass injection conducted by the colonial government. How well supported is this?

Yes, this is the leading theory on the origin of HIV and is well supported by current evidence.

One of the main tools here that has become dramatically more useful within the last two decades or so and is becoming exponentially more common due to improvements in technology is genetic sequencing of viruses. There's a lot of complexity here I'll gloss over but in general if you can sequence a viral genome you can then compare genetic distances to other viral samples and look for similarities. This makes it possible to track different strains or variants which you can use to reconstruct a "family tree" of sorts. And using this in concert with information about geographic locations and dates can make it possible to trace the spread of a virus.

For example, let's say you have some disease and in year 1 you can identify three different strains of it in the city of Old Towne, named X, Y, and Z. Then you find in year 2 there are not only strains X, Y, and Z but also U, V, and W, which are descendants of strain X. In year 5 you find that in another city, New Town, there are many samples of strain U. And in year 7 you find in New Town that there are now strains A, B, and C as well, all descendants of U. This tells a story of how strain U was imported from Old Towne to New Town and became the seed of local infections there. In general, places with greater local diversity of strains tend to be indicative of places where the disease has existed for longer.

And these are the patterns we see with HIV-1. By the 1970s and '80s it had started to spread globally. But in the 1960s there was a huge genetic diversity in HIV samples from samples from the 1960s in Kinshasa, that much has been known since the mid 2000s. More recently more thorough analysis of a larger number of samples combined with more sophisticated modeling has made it possible to reconstruct the spread of HIV within and out of Kinshasa (then known as Leopoldville) starting in the early 1920s (95% probability within the range 1909 to 1930).

One of the additional lines of evidence is that it's possible to identify strains of HIV that have spread locally or regionally near Kinshasa but have not spread globally. This can also be seen with the HIV-1 group O strains which appear to be much less infectious and have only spread locally, compared to the group M strains which make up the bulk of the pandemic strains of HIV. This is a similar story we've seen with other diseases where a more transmissible variant spreads more widely and becomes dominant as the major infectious strain globally.

In short, even less sophisticated analyses reveal the area near Kinshasa as the likely geographic origin of the HIV-1 pandemic at least as early as the 1960s, and more sophisticated computer aided modeling and phylogenetic mapping trace the origin with a high degree of confidence back to Kinshasa/Leopoldville in what was then the Belgian Congo in the 1920s plus or minus about a decade.