Why doesn't Bukovský's classification of forcing extensions solve Shelah's dream?

In Shelah's Logical Dreams, he poses the following problem:

Show that forcing is the unique method in a non-trivial sense.

In 1973 Bukovský proved (with a more recent reformulated proof, and a different proof by Friedman, Fuchino, and Sakai) that if $M \subseteq N$ is an extension of transitive models of ZFC with the same ordinals and $\kappa$ is a regular uncountable cardinal, then $N$ is a $\kappa$-c.c. generic extension iff every function $f \in N$ with $\mathrm{dom}(f) \in M$ and $\operatorname{rng}(f) \subseteq M$ there is a $g \in M$ with $\operatorname{dom}(g) = \operatorname{dom}(f)$ such that $M \models |g(i)| < \kappa$ for all $i \in \operatorname{dom}(f)$.

Why doesn't this characterization (along with the related semiset-formulated results) satisfactorily answer Shelah's question? Is there something more one could hope for?