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Detecting depression from speech in elderly patients with mild cognitive impairment (MCI) is complicated by overlapping acoustic effects of cognitive decline. Without disentangling these, classifiers risk learning cognitive rather than depression-specific patterns. We present a Korean elderly speech corpus of 89 MCI speakers collected over three years with concurrent depression (SGDS) and cognitive (MMSE) assessments. Both linear mixed-effects models and partial correlations identify the same pattern: of seven feature groups, only formant features are associated with depression after controlling for cognitive function, while widely used F0 features carry no signal. The F1+F2 subset achieves UAR 0.760 with just 12 features, outperforming both the full eGeMAPS baseline and self-supervised representations. These findings show that concurrent clinical assessments can identify interpretable, depression-specific acoustic markers in populations where cognitive and affective symptoms co-occur.