Please refer to Boyd et al.'s convergence analysis of ADMM (Chapter 3 and Appendix A). In Section 3.2.1, it says that the primal variables $x^k, z^k$ need not converge to optimal values.
Now, as discussed in Section 3.3, ADMM satisfies the dual feasibility condition $0 \in \partial g(z^*) + B^T y^*$ at every iteration, while $0 \in \partial f(x^*) + A^T y^*$ and primal feasibility $A x^* + B z^* - c = 0$ are satisfied in limit.
These 3 conditions are essentially the KKT conditions for the problem (3.1) - see page 13. Also, under strong duality (which is satisfied here - convex problem with affine equality constraint), all the primal-dual solution pairs satisfy KKT conditions (see http://www.stat.cmu.edu/~ryantibs/convexopt-F18/lectures/kkt.pdf). Since KKT conditions are being satisfied in the limit here, shouldn't both the primal and dual variables be optimal in the limit as well?
I understand that without additional assumptions, primal objective convergence for general convex functions need not lead to primal variable convergence. But, what am I missing in the reasoning above?
What you were missing is that, $(x^k)$ and $(z^k)$ may not converge at all, without further assumptions.