  In this paper we consider kinetically constrained models (KCM) on $\mathbb
Z^2$ with general update families $\mathcal U$. For $\mathcal U$ belonging to
the so-called "critical class" our focus is on the divergence of the infection
time of the origin for the equilibrium process as the density of the
facilitating sites vanishes. In a recent paper Mar\^ech\'e and two of the
present authors proved that if $\mathcal U$ has an infinite number of "stable
directions", then on a doubly logarithmic scale the above divergence is twice
the one in the corresponding $\mathcal U$-bootstrap percolation.
  Here we prove instead that, contrary to previous conjectures, in the
complementary case the two divergences are the same. In particular, we
establish the full universality partition for critical $\mathcal U$. The main
novel contribution is the identification of the leading mechanism governing the
motion of infected critical droplets. It consists of a peculiar hierarchical
combination of mesoscopic East-like motions.
