As social robots take on increasingly complex roles like game masters (GMs) in multi-party games, the expectation that physicality universally enhances user experience remains debated. This study challenges the "one-size-fits-all" view of tangible interaction by identifying a critical boundary condition: users' Negative Attitudes towards Robots (NARS). In a between-subjects experiment (N = 67), a custom-built robot GM facilitated a multi-party murder mystery game (MMG) by delivering clues either through direct tangible interaction or a digitally mediated interface. Baseline multivariate analysis (MANOVA) showed no significant main effect of delivery modality, confirming that tangibility alone does not guarantee superior engagement. However, primary analysis using multilevel linear models (MLM) revealed a reliable moderation: participants high in NARS experienced markedly lower narrative immersion under tangible delivery, whereas those with low NARS scores showed no such decrement. Qualitative findings further illuminate this divergence: tangibility provides novelty and engagement for some but imposes excessive proxemic friction for anxious users, for whom the digital interface acts as a protective social buffer. These results advance a conditional model of HRI and emphasize the necessity for adaptive systems that can tailor interaction modalities to user predispositions.