  We present a multi-variable extension of Rubio de Francia's restricted
weak-type extrapolation theory that does not involve Rubio de Francia's
iteration algorithm; instead, we rely on the following Sawyer-type inequality
for the weighted Hardy-Littlewood maximal operator $M_u$:
  $$ \left \Vert \frac{M_u (fv)}{v} \right \Vert_{L^{1,\infty}(uv)} \leq
C_{u,v} \Vert f \Vert_{L^1(uv)}, \quad u, \, uv \in A_{\infty}. $$
  Our approach can be adapted to recover weak-type $A_{\vec P}$ extrapolation
schemes, including an endpoint result that falls outside the classical theory.
  Among the applications of our work, we highlight extending outside the Banach
range the well-known equivalence between restricted weak-type and weak-type for
characteristic functions, and obtaining mixed and restricted weak-type bounds
with $A_{p}^{\mathcal R}$ weights for relevant families of multi-variable
operators, addressing the lack in the literature of these types of estimates.
We also reveal several standalone properties of the class $A_{p}^{\mathcal R}$.
