 Almost sure path regularity does not ensure progressive measurability

Let a probability space have two sample points, one of probability zero and one of probability one, and give it the complete power-set sigma-algebra at every time. Let $A\subseteq [0,1]$ be non-Borel. Assign the stochastic process the path $1_A(t)$ at the null sample point and the zero path at the other point. Each fixed-time variable is measurable and the paths are càdlàg almost surely, but the stochastic process is not progressively measurable: a section in the time variable of a product-measurable function must be Borel measurable at every sample point. The zero stochastic process is an indistinguishable progressively measurable version. Completing the filtration does not complete the time-product sigma-algebra or remove this distinction.