Prod Check Gadget in PLONK - any polynomial which satisfies the prod check seems to be the trivial polynomial

In Dan Boneh's PLONK Video - https://www.youtube.com/watch?v=vxyoPM2m7Yg he refers to the Prod Check Gadget

$\omega \in F_p$ is a primitive $k$-th root of unity ($\omega^{k-1} = 1$)

$H = \{1, \omega, \omega^2, ..., \omega^{k-1}\} \subseteq F_p$

Prod Check Gadget is used to prove that $\prod_{a \in H} f(a) = c$

I was trying to construct a polynomial $f(x)$ which there is true.

I used $p=17$ i.e $f \in F_{17}$

$\omega = 4$ is the primitive $4$th root of unity in $F_{17}$.

So $H = \{1, 4, 16, 13\}$

Let $c = 5$

Now, I try to find a polynomial $f$ such that $f(a) = 5$ for all $a \in H$ using Lagrange Polynomial Interpolation.

sage: F17 = GF(17)
sage: R17.<x> = PolynomialRing(F17)
sage: w = F17(4)
sage: c = F17(5)
sage: points = [(w^0, c), (w, c), (w^2, c), (w^3, c)]
sage: R17.lagrange_polynomial(points)
5

So Lagrange Interpolation gives returns $f$ as the trivial polynomial $f(x) = 5$

Any value of $c$ returns the polynomial as $f(x) = c$

So I am unable figure what is the point of this gadget?

Or am I doing something wrong in my toy example?