The primary function of the centromere during cell division is to serve as the attachment site for spindle fibers. During mitosis and meiosis, spindle fibers, which are composed of microtubules, connect to the centromere via a protein complex called the kinetochore. This connection is crucial because it allows the chromosomes to be accurately aligned and then segregated into the daughter cells. Without this attachment, chromosomes would not be pulled apart correctly, leading to unequal distribution of genetic material.

To clarify other options:
- The origin of DNA replication is a different specific DNA sequence, not located at the centromere.
- The site of ribosomal RNA synthesis is the nucleolus, not the centromere.
- While spindle checkpoint proteins monitor attachment at the centromere, the centromere itself is not their specific location; rather, these proteins function at or near the kinetochore complex.

Thus, the centromere's indispensable role is as the binding site for spindle fibers, which ensures proper chromosome separation.

Reference: Molecular Biology of the Cell, 6th Edition, Chapter 18: Mitosis and Cytokinesis