Rydberg atom arrays offer flexible geometries of strongly interacting neutral atoms, which are useful for many quantum applications such as quantum simulation and quantum computation. Here, we consider an all-optical gate-based quantum computing scheme for the Rydberg atom arrays, in which auxiliary atoms (wire atoms) are used as a mean of quantum-mechanical remote-couplings among data-qubit atoms, and optical individual-atom addressing of the data and wire atoms is used to construct universal quantum gates of the data atoms. The working principle of our gates is to use the wire atoms for coupling mediation only, while leaving them in noncoupling ground states before and after each gate operation, which allows the double-excited states of data qubits to be accessible by a sequence of π or π/2 pulses addressing the data and wire atoms. Optical pulse sequences are constructed for standard one-, two-, and multi-qubit gates, and the arbitrary two-qubit state preparation is considered for universal computation prospects. We further provide a detailed resource estimate for an experimental implementation of this scheme in a Rydberg quantum simulator.