Why don't Antarctic waters have more upwellings, when Arctic waters are so rich in upwellings?
First, we might want to distinguish between the different types of upwelling:


Coastal upwelling: It is the best known form of upwelling. Winds in these systems flow parallel to the coast (with the coast to the left in the northern hemisphere or to the right in the southern hemisphere) and generate upwelling dynamics. Surface Ekman balance is setup (in deep enough waters) with water transport being to the right (left) of the wind in the northern (southern) hemisphere. The result is a deficit along the coast that requires a compensating flow in the deeper part of the water column bringing usually colder waters to the surface. If the wind blows in the opposite direction, then downwelling occurs.
The figure from the question refers to coastal upwelling (here in its original website with explaining caption). There is definitely more coastal upwelling in the northern hemisphere because of the abundance of shorelines as a greater amount of landmass is present in that hemisphere. 


 Source Commons Wikipedia.


Equatorial upwelling: Winds from the east blow and converge along the equator as part of the Intertropical Convergence Zone (ITCZ). While the magnitude of the Coriolis acceleration is zero along the equator, Ekman transport takes place immediately north and south of the equator resulting in surface ocean divergence that requires a compensating flow of denser (nutrient-rich) water upwelled from below.


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Southern Ocean upwelling: Strong westerly (eastward) winds dominate the atmospheric setting around Antarctica. The strong winds force the Antarctic Circumpolar Current, the strongest current in the ocean. The winds also result in an Ekman balance with northward flow in the upper part of the water column. To compensate that flow, some water flows up from greater depths. The magnitude of the Southern Ocean upwelling is still being studied, but some observations and models propose that it represents the main way for dense water from deeper areas to be reintroduced in the surface circulation. If that were the case, then it represents the largest upwelling system in the world. An alternative explanation proposes oceanic eddies as the main mechanism for dense water to reconnect with the surface circulation. Additionally, there is coastal upwelling over the Antarctic continental shelf driven by local winds that brings relatively warmer waters to the shelf contributing to local/regional ice melting.
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The question also mentions downwelling associated with dense water formation. Deep water forms at high latitudes (typically north and south of 60°) in both the southern and northern hemispheres. As mentioned in the question, ice contains almost no salt, so when ice forms, a salty, sub-zero brine is left behind and sinks. In the Antarctic, as the saltier water moves down, it mixes with slightly warmer intermediate water. The mixture becomes Antarctic Bottom Water (AABW), which flows northwards along the bottom in all the main oceans. In the Arctic, the northward flowing branch of the Gulf Stream (salty and warm) joins the subpolar gyre and enters the Labrador Sea where as it cools, it starts to sink. The eastern branch of the Gulf Stream begins to sink in the Nordic Seas. The resulting water masses combine to form the dense North Atlantic Deep Water (NADW) that flows south at depth. Both the creation of AABW and NADW can be considered downwelling, as there is a net downward flow in the formation areas, but they are not related to wind-driven downwelling in the vicinity of coasts.