Near-field blockage changes the beam-design objective in millimeter-wave links: maximizing the unblocked on-axis gain does not necessarily maximize blocked-link performance. This paper studies when phase-only, aperture-constrained non-diffracting (ND) beams provide a blocked-link advantage over equal-aperture, equal-power conventional reference beams. We develop a unified annular-spectrum framework that generates isotropic Bessel-like and anisotropic Mathieu-like beams under discrete phased-array constraints, and a geometry-aware analysis centered on three propagation landmarks: the peak-intensity distance, the crossover distance, and an effective post-blockage recovery distance. Their relationship yields a recovery-before-crossover condition linking blockage size, depth, cone angle, and usable ND range, and motivates a blocked-link gain ratio that maps directly onto an achievable-rate gap at every operating SNR. The analysis also explains why anisotropic Mathieu-like beams can outperform isotropic ones under direction-dependent blockage. Monte Carlo simulations verify the predicted advantage regimes, an auxiliary comparison against a near-field focusing baseline confirms that the advantage persists against an unblocked-optimal array, and sensitivity studies over cone-angle choice and partial-transmission blockers show that the opaque-screen picture is a conservative reading of the underlying physics. The results identify Bessel-like and Mathieu-like beams as practical candidates for blockage-resilient near-field communications.