We propose a measurement-based FTQC (MB-FTQC) architecture for high-connectivity platforms such as trapped ions and neutral atoms. The key idea is to use verified logical ancillas combined with Knill’s error-correcting teleportation, eliminating repeated syndrome measurements and simplifying decoding to logical Pauli corrections, thus keeping classical overhead low. To align with near-term device scales, we present two implementations benchmarked under circuit-level depolarizing noise: (i) a Steane-code version that uses analog RZ (θ) rotations, akin to the STAR architecture [Akahoshi et al., PRX Quantum 5, 010337], aiming for the megaquop regime (∼ 10⁶ T gates) on devices with thousands of qubits; and (ii) a Golay-code version with higher-order zero-level magic-state distillation, targeting the gigaquop regime (∼ 10⁹ T gates) on devices with tens of thousands of qubits. At a physical error rate p = 10⁻⁴, the Steane path supports 5 × 10⁴ logical RZ (θ) rotations, corresponding to ∼ 2.4 × 10⁶ T gates and enabling megaquop-scale computation. With about 2,240 physical qubits, it achieves log2 QV = 64. The Golay path supports more than 2 × 10⁹ T gates, enabling gigaquop-scale computation. These results suggest that our architecture can deliver practical large-scale quantum computation on near-term high-connectivity hardware without relying on resource-intensive surface codes or complex code concatenation.