High-value greenhouse crops require quality-sensitive selective harvesting under labor shortages and variable crop conditions. This review synthesizes 188 unique primary studies and examines robotic harvesting as a complete task chain rather than a set of isolated sensing or manipulation modules. Embodied intelligence is used as an analytical lens to connect perception, harvestability assessment, decision making, manipulation, feedback, failure propagation, and recovery. A greenhouse-cucumber case study illustrates how errors propagate across these stages. The synthesis shows that the dominant bottleneck is system integration: incomplete observability, short-lived scene states, contact uncertainty, and weak outcome verification or recovery cause cumulative losses in complete-task success, effective throughput, and product quality. Most reported systems remain at closed-loop automation or early embodied interaction, with limited evidence of interaction-driven learning or cross-context generalization. We therefore frame evaluation around six production-relevant dimensions: complete-task success, effective cycle time and retry cost, product quality, failure detection and recovery, human intervention, and continuous operation. Near-term priorities are reliable, recoverable operation in defined crop–facility systems and standardized reporting; adaptation across cultivars and greenhouse configurations is a medium-term objective, whereas embodied learning and broad cross-platform transfer remain longer-term research directions. This system-level perspective reframes smart robotic harvesting around information continuity, fault containment, and deployable greenhouse performance.