We experimentally optimize the frequency of flux-tunable couplers in a superconducting quantum processor to minimize the impact of spectator transmons during quantum operations (single-qubit gates, two-qubit gates, and readout) on other transmons. We adapt a popular transmonlike tunable-coupling element, achieving high-fidelity, low-leakage controlled-Z gates with unipolar, fast-adiabatic pulsing only on the coupler. We demonstrate the ability of the tunable coupler to null residual-ZZ coupling as well as exchange couplings in the one- and two-excitation manifolds. However, the nulling of these coherent interactions is not simultaneous, prompting the exploration of trade-offs. We present experiments pinpointing spectator effects on specific quantum operations. We also study the combined effect on the three types of operations using repeated quantum parity measurements.