Falls are a major cause of injury and mortality among older adults, yet most incidents occur in private indoor environments where monitoring must balance effectiveness with privacy. Existing privacy-preserving fall detection approaches, particularly those based on radio frequency sensing, often rely on coarse motion cues, which limits reliability in real-world deployments. We introduce TaFall, a balance-informed fall detection system based on low-cost, privacy-preserving thermal array sensing. The key insight is that TaFall models a fall as a process of balance degradation and detects falls by estimating pose-driven biomechanical balance dynamics. To enable this capability from low-resolution thermal array maps, we propose (i) an appearance-motion fusion model for robust pose reconstruction, (ii) physically grounded balance-aware learning, and (iii) pose-bridged pretraining to improve robustness. TaFall achieves a detection rate of 98.26% with a false alarm rate of 0.65% on our dataset with over 3,000 fall instances from 35 participants across diverse indoor environments. In 27 day deployments across four homes, TaFall attains an ultra-low false alarm rate of 0.00126% and a pilot bathroom study confirms robustness under moisture and thermal interference. Together, these results establish TaFall as a reliable and privacy-preserving approach to fall detection in everyday living environments.