Does there always exist equivalent (M)(I)LPs with and without objective functions?
Yes (assuming you truly did mean "if" rather than "if and only if").

Add any objective function you want (e.g., 1, or the sum of the variables, or anything), then feed it to a MILP solver and see whether it returns 'Feasible' or 'Infeasible'.  Then the resulting MILP is feasible if and only if the original MILP was feasible.  Any optimal solution to the resulting MILP is feasible and thus also a feasible solution to the original MILP problem, and consequently represents a valid pure Nash equilibrium.