Was a quotient of two norms considered as a constraint to a convex optimization problem before?

As @Mark L. Stone commented, that constraint isn’t convex (and therefore not a convex optimization problem). You could instead consider the different constraint:
$$|x|_{infty} leq sM$$
$$|x|_{2} leq M$$
which is convex. Note that the elements $x$ satisfying $|x|_{infty} leq s |x|_2$ and $|x|_2 leq M$ satisfy this constraint. So the solution $tilde{x}^*$ of this new optimization problem will do at least as well (in terms of the criteria $g$) as the solution $x^*$ of your nonconvex optimization problem as long as $|x^*|_{2} leq M$ (of course, you might be overfitting).

From a practical perspective, I can’t imagine there is any loss in using this constraint. Also, this type of constraint has been used in practice (e.g. Lasso/Ridge/ElasticNet regression with a lower/upper bound on the coefficients, bounded Lipschitz regression).