We explore a small quantum refrigerator in which the working substance is made of paradigmatic nearest-neighbor quantum spin models, the XYZ and the XY model with Dzyaloshinskii-Moriya interactions, consisting of two and three spins, each of which is in contact with a bosonic bath. We identify a specific range of interaction strengths which can be tuned appropriately to ensure a cooling of the selected spin in terms of its local temperature in the weak-coupling limit. Moreover, we report that in this domain, when one of the interaction strengths is disordered, the performance of the thermal machine operating as a refrigerator remains almost unchanged instead of degrading, thereby establishing the flexibility of this device. However, to obtain a significant amount of cooling via ordered as well as disordered spin models, we observe that one has to go beyond the weak-coupling limit and compute the figures of merit by using global master equations.