The standard max-min inequality states that, for a bivariate function $f: X \times Y \to \mathbb{R}$,
$$ \sup_x \inf_y f(x,y) \le \inf_y \sup_x f(x,y). \qquad (*) $$
I wonder whether the following property is also a max-min inequality:
$$ \sup_x \inf_y f(x,y) \le \inf_x \sup_y f(x,y), \qquad (**) $$
since $$ \begin{align} \inf_{y_0} f(x,y_0) &\le f(x,y) &\quad \forall x\forall y\\ \sup_{x_0} \inf_{y_0} f(x_0,y_0) &\le \sup_{x_0} f(x_0,y) &\quad \forall y\\ \sup_{x_0} \inf_{y_0} f(x_0,y_0) &\le \inf_{y_0}\sup_{x_0} f(x_0,y_0). \end{align} $$
I am wondering whether (a) the proposed min-max inequality (**) is indeed correct based on the proof, and if so, (b) why it is not as widely used or referenced in minimax theory.
The proof you gave seems to conclude with (*) rather than (**).
(**) is false, as you can see for instance with $f(x,y)=x$.