Why is the concentration of highly siderophile elements so uniform in the mantle?
First, note that this plot is normalized to a reference, here the composition of CI chondrites (your version lacks the Y axis label, it's figure 2.1 from Walker 2016). So if the concentration of highly siderophile elements (HSE) seems uniform, it's partly because it is a relative concentration, not an absolute one. If you look at the raw data (Table 1.1), you'll see that the absolute concentrations vary by more than one order of magnitude depending on the element (from 0.35 ppb for Re to 7.6 for Pt). Also, as you noted the scale is logarithmic, which tends to flatten the differences.

Now, why do HSE have the same relative abundance in the mantle compared to chondrites? Because elements from the same family show the same behavior—actually it's why they form a family in the first place. HSE all are, well, highly siderophile, so they all went into the core during the core segregation, and now they all are highly depleted in the mantle compared to chondrites.

Note that despite this depletion, HSE are actually still too abundant in the mantle: there should be even less of them. Geochemists have long debated this issue. This nice paper (Walker 2009) reviews the different hypotheses that might explain this HSE "enrichment": incomplete core segregation, lower metal/silicate partitioning, late accretion of chondritic material after core segregation...