We propose a directional pinching-antenna system (DiPASS), a comprehensive framework that transitions PASS modeling from idealized abstraction to physical consistency. DiPASS introduces the first channel model that accurately captures the directional, pencil-like radiation of pinching antennas, incorporates a practical waveguide attenuation of 1.3 dB/m, and accounts for stochastic line-of-sight blockage. A key enabler of DiPASS is our new "equal quota division" power allocation strategy, which guarantees predetermined coupling lengths independent of antenna positions, thereby overcoming a critical barrier to practical deployment. Our analysis yields foundational insights: we derive closed-form solutions for optimal antenna placement and orientation in single-PA scenarios, quantifying the core trade-off between waveguide and free-space losses. For multi-PA systems, we develop a scalable optimization framework that leverages directional sparsity, revealing that waveguide diversity surpasses antenna density in enhancing system capacity. Extensive simulations validate our analysis and demonstrate that DiPASS provides a realistic performance benchmark, fundamentally reshaping the understanding and design principles for future PASS-enabled 6G networks.