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).