Some deep learning-based point cloud registration methods struggle with zero-shot generalization, often requiring dataset-specific hyperparameter tuning or retraining for new environments. We identify three critical limitations: (a) fixed user-defined parameters (e.g., voxel size, search radius) that fail to generalize across varying scales, (b) learned keypoint detectors exhibit poor cross-domain transferability, and (c) absolute coordinates amplify scale mismatches between datasets. To address these three issues, we present BUFFER-X, a training-free registration framework that achieves zero-shot generalization through: (a) geometric bootstrapping for automatic hyperparameter estimation, (b) distribution-aware farthest point sampling to replace learned detectors, and (c) patch-level coordinate normalization to ensure scale consistency. Our approach employs hierarchical multi-scale matching to extract correspondences across local, middle, and global receptive fields, enabling robust registration in diverse environments. For efficiency-critical applications, we introduce BUFFER-X-Lite, which reduces total computation time by 43% (relative to BUFFER-X) through early exit strategies and fast pose solvers while preserving accuracy. We evaluate on a comprehensive benchmark comprising 12 datasets spanning object-scale, indoor, and outdoor scenes, including cross-sensor registration between heterogeneous LiDAR configurations. Results demonstrate that our approach generalizes effectively without manual tuning or prior knowledge of test domains. Code: https://github.com/MIT-SPARK/BUFFER-X.