Can I just use a stain-blocking primer on basement mold?

Primer is a paint, and paint is the last step. On active growth or a damp wall it blisters. On a stone foundation it can slow the drying you actually need. After removal and a dry meter reading, a primer may be a fine cosmetic layer on new drywall. It is not a treatment for the old paper still in the wall.

If a contractor's whole plan is 'Kilz it,' get another scope before you pay. The product is not the problem. Using it as a substitute for drying is.