Automated Alzheimer's Disease (AD) screening has predominantly followed the inductive paradigm of pattern recognition, which directly maps the input signal to the outcome label. This paradigm sacrifices construct validity of clinical protocol for statistical shortcuts. This paper proposes Agentic Cognitive Profiling (ACP), an agentic framework that realigns automated screening with clinical protocol logic across multiple cognitive domains. Rather than learning opaque mappings from transcripts to labels, the framework decomposes standardized assessments into atomic cognitive tasks and orchestrates specialized LLM agents to extract verifiable scoring primitives. Central to our design is decoupling semantic understanding from measurement by delegating all quantification to deterministic function calling, thereby mitigating hallucination and restoring construct validity. Unlike popular datasets that typically comprise around a hundred participants under a single task, we evaluate on a clinically-annotated corpus of 402 participants across eight structured cognitive tasks spanning multiple cognitive domains. The framework achieves 90.5% score match rate in task examination and 85.3% accuracy in AD prediction, surpassing popular baselines while generating interpretable cognitive profiles grounded in behavioral evidence. This work demonstrates that construct validity and predictive performance need not be traded off, charting a path toward AD screening systems that explain rather than merely predict.