Strong algebraic proof systems such as IPS (Ideal Proof System; Grochow-Pitassi 2018) offer a general model for deriving polynomials in an ideal and refuting unsatisfiable propositional formulas, subsuming most standard propositional proof systems. A major approach for lower bounding the size of IPS refutations is the Functional Lower Bound Method (Forbes, Shpilka, Tzameret and Wigderson 2021), which reduces the hardness of refuting a polynomial equation f(x) = 0 with no Boolean solutions to the hardness of computing the function 1/f(x) over the Boolean cube with an algebraic circuit. Using symmetry we provide a general way to obtain many new hard instances against fragments of IPS via the functional lower bound method. This includes hardness over finite fields and hard instances different from Subset Sum variants, both of which were unknown before, and significantly improved constant-depth IPS lower bounds. Conversely, we expose the limitation of this method by showing it cannot lead to proof complexity lower bounds for any hard Boolean instance (e.g., CNFs) for any sufficiently strong proof systems (including AC0[p]-Frege).