Why is the South Pole Telescope located exactly at the South Pole?

Just guessing here, but ...

Compare the regions with really good skies with the places that have infrastructure and people present. Most of the installations are coastal, right? Are those good places to put a telescope? And while the whole inland plateau has good skies, it has few occupied site, and only one operated by the US.

So what is the case for putting up some other (very expensive to build and maintain) installation, when you could just drop it by South Pole Station where they already maintain a year-round presence.

Here are extra reasons to the dry air :

During the winter, sunshine does not reach the South Pole; nighttime (or daytime in the summer) extends for months. The lack of daily sunsets and sunrises makes the atmosphere extremely stable. Conducting observations in the winter also removes another contaminant to millimeter/sub-millimeter observations - the sun. All these factors conspire to make the South Pole the perfect place for the South Pole Telescope.

The further north these extra reasons for choosing the South Pole plateau become important, there will be increasing presence of the sun.

If the telescope was situated directly on the southern axis of the earth's rotation, the telescope's declination axis would be at zenith. The base for the axis would be level to the ground. In theory you could compensate for the earth's rotation with only one motion of the telescope. Also, its the only place on earth where the entire southern celestial hemisphere is visible. Now, are these the reasons it is built there, probably not but they would be advantages.

There is no scientific reason that the South Pole Telescope is located so close (within a few 100 meters, really) of the geographic south pole. It might as well be located a few hundred miles to either direction, no difference.

The reason is purely practical: On the South Pole, there exists very significant infrastructure in the form of the permanently staffed Amundsen-Scott South Pole Station, including power generators, an airfield, telecommunication channels to get the data off to warmer offices ;) and of course permanent scientific and technical staff.

Alternative permanent research areas are mostly near the coast of Antarctica, and there indeed the seeing would be much worse. The two exceptions are the Russian Vostok station and the French/Italian Concordia station, but the South Pole Telescope is mostly US-funded. What is more, there are dedicated pots of funding available for research in this location, as opposed to e.g. some high-altitude site in Chile.

Without a real scientific reason, it is clear that the question of why the Amundsen-Scott station is exactly at the pole is not a scientific one, but is geo-political in the context of the emerging cold war in the 1950s.

That said, the extremely clean and dry air and the high altitude (almost 3km) at the south pole are crucial for a millimeter-telescope. But this location also means that the telescope has always the same sky above. That would be a show-stopper for a survey telescope, but is an advantage if you want to collect detailed ("deep") information about whatever area in front of you. Also it is nice when building a telescope since a simple azimuthal mount at the pole is in fact an equatorial mount there.

Finally, there is a long history of telescopes and instruments at the south pole for exactly these reasons. That in turn means that any new telescope, such as the South Pole Telescope, can now capitalize on a well-studied site, and can observe the same patches of sky to build on previous observations.