A similar idea has been used in the paper A Hierarchy of Relaxations between the Continuous and Convex Hull Representations for Zero-One Programming Problems by Sherali and Adams (1990).
From the abstract (emphasis mine):
In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. (...)
That is, equalities like the one you propose can be used to derive stronger linear programming relaxations, up to the convex hull for zero-one programming problems.
The authors of the above paper also mention applications in generating strong or facetial valid inequalities.