Modern HTTPS mechanisms such as Encrypted Client Hello (ECH) and encrypted DNS improve privacy but remain vulnerable to website fingerprinting (WF) attacks, where adversaries infer visited sites from encrypted traffic patterns. Existing WF methods rely on supervised learning with site-specific labeled traces, which limits scalability and fails to handle previously unseen websites. We address these limitations by reformulating WF as a zero-shot cross-modal retrieval problem and introducing STAR. STAR learns a joint embedding space for encrypted traffic traces and crawl-time logic profiles using a dual-encoder architecture. Trained on 150K automatically collected traffic-logic pairs with contrastive and consistency objectives and structure-aware augmentation, STAR retrieves the most semantically aligned profile for a trace without requiring target-side traffic during training. Experiments on 1,600 unseen websites show that STAR achieves 87.9 percent top-1 accuracy and 0.963 AUC in open-world detection, outperforming supervised and few-shot baselines. Adding an adapter with only four labeled traces per site further boosts top-5 accuracy to 98.8 percent. Our analysis reveals intrinsic semantic-traffic alignment in modern web protocols, identifying semantic leakage as the dominant privacy risk in encrypted HTTPS traffic. We release STAR's datasets and code to support reproducibility and future research.