Kernel Support Vector Machines (KSVMs) are effective methods for nonlinear classification but face scalability challenges due to high training costs. We explore kernel thinning (KT) as a coreset selection method for KSVMs, focusing on its impact on performance, computational efficiency, and its effect and connection with support vectors. Our experiments show that KT-based models achieve comparable accuracy to full KSVMs while using significantly fewer training samples and iterations. Notably, the KT coresets do not strongly overlap with traditional support vectors, suggesting a distinct yet effective representation. We also demonstrate that KT enables fast, reliable hyperparameter tuning, making it a practical approach for scalable SVM kernel learning.