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Noisy, Intermediate Scale Quantum (NISQ) computers have reached the point where they can show the potential for quantum advantage over classical computing. Unfortunately, NISQ machines introduce sufficient noise that even for moderate size quantum circuits the results can be unreliable. We propose a co-designed superconducting quantum computer using a Superconducting Nonlinear Asymmetric Inductive eLement (SNAIL) modulator. The SNAIL modulator is designed by considering both the ideal fundamental qubit gate operation while maximizing the qubit coupling capabilities. First, the SNAIL natively implements $\sqrt[n]{\texttt{iSWAP}}$ gates realized through proportionally scaled pulse lengths. This naturally includes $\sqrt{\texttt{iSWAP}}$, which provides an advantage over $\texttt{CNOT}$ as a basis gate. Second, the SNAIL enables high-degree couplings that allow rich and highly parallel qubit connection topologies without suffering from frequency crowding. Building on our previously demonstrated SNAIL-based quantum state router we propose a quantum 4-ary tree and a hypercube inspired corral built from interconnected quantum modules. We compare their advantage in data movement based on necessary \texttt{SWAP} gates to the traditional lattice and heavy-hex lattice used in latest commercial quantum computers. We demonstrate the co-design advantage of our SNAIL-based machine with $\sqrt{\texttt{iSWAP}}$ basis gates and rich topologies against $\texttt{CNOT}$/heavy-hex and $\texttt{FSIM}$/lattice for 16-20 qubit and extrapolated designs circa 80 qubit architectures. We compare total circuit time and total gate count to understand fidelity for systems dominated by decoherence and control imperfections, respectively. Finally, we provide a gate duration sensitivity study on further decreasing the SNAIL pulse length to realize $\sqrt[n]{\texttt{iSWAP}}$ qubit systems to reduce decoherence times.