Designing effective Human-Robot Interaction in task-oriented settings requires carefully balancing user engagement with socially acceptable levels of robot intrusiveness. In this paper, we examine how different robot intervention strategies shape user experience, interaction dynamics, perceived intrusiveness, and sense of support. We compare two approaches: a continuous engagement-seeking robot strategy, and a context-aware strategy that selectively intervenes based on the user's state and task context. Both approaches rely on multimodal behavioral cues, including body orientation and attentional signals, to guide robot actions. We evaluate these strategies in a user study with 32 participants performing a task in a simulated hospital environment. Our findings show that higher interaction frequency does not necessarily lead to better engagement. Instead, we observe a systematic trade-off between perceived support and intrusiveness, influenced by factors such as physical proximity and user effort. These results provide empirical evidence that effective engagement in HRI depends on adaptive, context-sensitive intervention policies.

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