Increasing connectivity and decreasing qubit-state delocalization without compromising the speed and accuracy of elementary gate operations are topical challenges in the development of large-scale superconducting quantum computers. In this theoretical work, we study a special honeycomb qubit lattice where each qubit inside a unit cell is coupled to every other one via two dedicated tunable couplers and a common central element. This results in an effective multi-mode interaction enabling tunable, on-demand, all-to-all connectivity between each qubit pair within the unit cell. We provide a thorough analysis of the unit cell, including a proposal for a novel and efficient conditional-Z gate scheme which takes advantage of the effective multi-mode coupling. We develop an experimentally viable pulse protocol for a single-step gate implementation which considerably improves the gate speed compared to the previous two-qubit-gate realizations suggested for architectures utilizing a center mode. We also show numerical results on how the presence of spectator qubits affects the average two-qubit-gate fidelity, and analyse how the multi-mode coupling structure mitigates the delocalization-induced crosstalk during simultaneous single-qubit gates within the unit cell. We also provide analytical estimates for the errors caused by relaxation and dephasing during a two-qubit-gate operation, including noise terms for the multi-mode coupling structure. Our multi-mode coupling architecture results in a good balance between increased connectivity and available parallelism, especially when several interacting unit cells form a quantum processing unit. We anticipate that the obtained results pave the way towards high-connectivity quantum processors with efficient and low-overhead quantum algorithms.