We present a hardware-native gadget framework for solving constraint satisfaction problems on Rydberg quantum computing architectures. Our approach introduces a compact $xo{r}_1$ gadget that enforces exactly-one constraints, ubiquitous in combinatorial optimization, directly through geometric embedding and blockade interactions. A key advantage of the $xo{r}_1$ gadget is its fixed, problem-size-independent detuning requirements: enforcing constraints through blockade interactions eliminates the need for large penalty terms, thereby substantially reducing the detuning range compared to Quadratic Unconstrained Binary Optimization (QUBO) formulations and improving experimental feasibility. By tailoring the construction to the geometric connectivity of Rydberg atom arrays, the framework bypasses the all-to-all physical couplings often assumed in logical encodings. This enables embeddings compatible with planar layouts and avoids highly connected arrangements. We develop scalable implementations that reduce atom count and connectivity overhead while avoiding extensive classical preprocessing, making them compatible with near-term neutral-atom hardware. As illustrations, we apply our framework to the gate-assignment and N-queens problems, highlighting its practicality, resource efficiency, and hardware compatibility. In these examples, we observe reductions in detuning range of up to 99% and savings in atom count and connectivity overhead of up to 54% compared to the QUBO method. These results establish a route toward implementing large-scale combinatorial optimization on Rydberg platforms beyond the limits of existing encodings.