Atomic systems, ranging from trapped ions to ultracold and Rydberg atoms, offer unprecedented control over both internal and external degrees of freedom at the single-particle level. They are considered among the foremost candidates for realizing quantum simulation and computation platforms that can outperform classical computers at specific tasks. In this work, a realistic experimental toolbox for quantum information processing with neutral alkaline-earth-like atoms in optical tweezer arrays is described. In particular, a comprehensive and scalable architecture based on a programmable array of alkaline-earth-like atoms is proposed, exploiting their electronic clock states as a precise and robust auxiliary degree of freedom, and thus allowing for efficient all-optical one- and two-qubit operations between nuclear spin qubits. The proposed platform promises excellent performance thanks to high-fidelity register initialization, rapid spin-exchange gates, and error detection in read-out. As a benchmark and application example, the expected fidelity of an increasing number of subsequent SWAP gates for optimal parameters is computed, which can be used to distribute entanglement between remote atoms within the array.