Localizing user-queried events through natural language is crucial for video understanding models. Recent methods predominantly adapt Video LLMs to generate event boundary timestamps to handle temporal localization tasks, which struggle to leverage LLMs' powerful semantic understanding. In this work, we introduce MeCo, a novel timestamp-free framework that enables video LLMs to fully harness their intrinsic semantic capabilities for temporal localization tasks. Rather than outputting boundary timestamps, MeCo partitions videos into holistic event and transition segments based on the proposed structural token generation and grounding pipeline, derived from video LLMs' temporal structure understanding capability. We further propose a query-focused captioning task that compels the LLM to extract fine-grained, event-specific details, bridging the gap between localization and higher-level semantics and enhancing localization performance. Extensive experiments on diverse temporal localization tasks show that MeCo consistently outperforms boundary-centric methods, underscoring the benefits of a semantic-driven approach for temporal localization with video LLMs.