Depression and anxiety in older adults with Mild Cognitive Impairment (MCI) are frequently underdiagnosed due to limited access to care. Multimodal analysis of remote clinical interviews is a scalable screening approach, but existing methods have three limitations. First, they do not correct temporal misalignment across multimodal features extracted at different resolutions, inducing spurious cross-modal associations. Second, remote recordings exhibit uneven modality dropout, but missing values are often zero-filled, making them indistinguishable from valid near-zero measurements. Finally, they do not jointly attribute predictions to modalities, questions, and interview moments, limiting fine-grained clinical interpretation. We propose a Temporally-Aligned, Missingness-Aware, Interpretable (TAMI) multimodal fusion framework. TAMI aligns speech, language, facial, and physiological features within question-answer segments on a shared timeline, encodes modality-level missingness over time, and conditions fusion on question context. In interviews with 49 older adults with MCI, TAMI achieved area under the receiver operating characteristic curve (AUROC) scores of 0.68 (depression) and 0.69 (anxiety). Fine-grained temporal alignment of multimodal features produced the largest performance gain (
Δ≥0.1). Multi-level interpretability analysis revealed that depression classification relied on eyegaze and open-ended questions, while anxiety classification depended on eyegaze and head pose, with attribution uniformly distributed across questions. Using only responses to the open-ended questions (5.1min), the depression model achieved an AUROC score of 0.67, which was not significantly different from using the full interview (19min) (
p>0.05). Our findings support designing interview protocols centered on open-ended questions for depression screening in older adults with MCI.