Erasure qubits are a promising platform for implementing hardware-efficient quantum error correction. Realizing the error-correction advantages of this encoding requires frequent mid-circuit erasure checks that are fast, high-fidelity, and scalable. Here, we realize erasure detection with a hardware-efficient circuit consisting of a single readout resonator dispersively and symmetrically coupled to both transmons of a dual-rail qubit. We use this circuit to demonstrate single-shot erasure detection in 384 ns with minimal impact on the dual-rail logical manifold, achieving a residual error per check of $6.0(2)\times {10}^{-4}$, with only $8(3)\times {10}^{-5}$ induced dephasing per check, and an erasure error per check of $2.54(1)\times {10}^{-2}$. The high degree of matched dispersive readout coupling ($\chi$-matching) within the dual-rail qubit code space also allows us to realize a new modality: time-continuous erasure detection performed in parallel with single-qubit gates. Here we achieve a median $7.2\times {10}^{-5}$ error per gate with $<1\times {10}^{-5}$ error induced by erasure detection. This demonstrates a reduction in erasure detection overhead as well as a crucial ingredient for soft information quantum error correction. Together, these results establish symmetrically coupled dispersive readout as a fast, hardware-efficient, and scalable component for erasure-based quantum error correction using transmon dual-rail qubits.