Existing Wi-Fi sensing systems rely on injecting high-rate probing packets to extract channel state information (CSI), leading to communication degradation and poor deployability. Although Integrated Sensing and Communication (ISAC) is a promising direction, existing solutions still rely on auxiliary packet injection because they exploit only CSI from data frames. We present UniFi, the first Wi-Fi-based ISAC framework that fully eliminates intrusive packet injection by directly exploiting irregularly sampled CSI from diverse communication packets across multiple frequency bands. UniFi integrates a CSI sanitization pipeline to harmonize heterogeneous packets and remove burst-induced redundancy, together with a time-aware attention model that learns directly from non-uniform CSI sequences without resampling. We further introduce CommCSI-HAR, the first dataset with irregularly sampled CSI from real-world dual-band communication traffic. Extensive evaluations on this dataset and four public benchmarks show that UniFi achieves state-of-the-art accuracy with a compact model size, while fully preserving communication throughput.