Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input
The $f$ you defined does not have this property when $n\ge 4$.
We can show that, if $(x_1,\dots,x_n)$ are arbitrary, when $$t_\sigma=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$$ then the set of images $T=\{t_\sigma:\sigma\text{ is a permutation}\}$ has maximum cardinality exactly $(n-1)!$.
Proof:
Write $f(x_1,\dots,x_n) = g(x_1,\dots,x_n)^n$. Note that a cyclic permutation of $(x_1,\dots,x_n)$ multiplies $g$ by $\alpha_n^k$ and therefore $f$ is multiplied by $\alpha_n^{kn} = 1$. So since $t_{\pi\circ\sigma}=t_\sigma$ when $\pi$ is one of the $n$ cyclic permutations, the maximum cardinality of $T$ is at most $n!/n=(n-1)!$.
Now, since $g$ is a linear function with distinct coefficients before each $x_i$, if you are working in a characteristic zero field, the set of all $g(x_{\sigma(1)},\dots,x_{\sigma(n)})$ has maximum cardinality $n!$. But since the equation $t_\sigma=X^n$ has at most $n$ solutions in $X$, the maximum cardinality of $T$ is at least $(n-1)!$, finishing the proof.
Now, why doesn't it work? In fact I don't know if the resolvent can be defined by a systematic formula that extends for all $n$: we just find ad-hoc polynomials that happen to let us reduce the equation to a degree as low as possible. As a result, proving that such a polynomial exists is much easier than proving that no polynomial exists, and I don't know if there are proofs of Abel-Ruffini that don't use group theory.
So, how could we define a resolvent for $n=4$? Since $n$ is even there are $\tbinom{n}{n/2}/2=3$ ways to partition the roots into a set of two equal-sized subsets, so that: $$f(x)=x_1x_2+x_3x_4$$ has at most 3 images under permutation of $x$ (and it is easy to check that this bound is reached). There are 8 symmetries that can be derived from the two generator equations: $$f(a,b,c,d)=f(b,a,c,d)\\ f(a,b,c,d)=f(c,d,a,b)$$ Note that in general $f(a,b,c,d)\ne f(b,c,d,a)$, so unlike when $n=3$ cylic permutations are not always symmetries. Also note that this construction is only really useful for $n=4$, since for $n=6$ we get $\tbinom{n}{n/2}/2=10>6$.
You can check Lagrange's original 1770 paper (in French) to see that it is indeed the way he defined cubic resolvents and quartic resolvents.
He also gives a much more complicated polynomial for quintics, which he derives by generalizing to any root of unity $\alpha$ ($\alpha^5=1$), expanding the $f$ you just defined as a linear combination of $\alpha^0,\dots,\alpha^{n-1}$ and taking only the coefficient of $\alpha^0$. It turns out the resulting expression is highly symmetric, and the 20 symmetries can be derived from the three generators: $$\begin{align} f(a,b,c,d,e)&=f(b,c,d,e,a)&&\text{cyclic permutation}\\ f(a,b,c,d,e)&=f(e,d,c,b,a)&&\text{reversal}\\ f(a,b,c,d,e)&=f(a,c,e,b,d)&&\text{doubling modulo 5} \end{align}$$ so that there are $5!/20=6$ distinct values. Cyclic permutation is for the same reason as when $n=3$; reversal and doubling modulo 5 come from the fact when replacing $\alpha$ by another root of unity $\alpha'=\alpha^k$ (where $k$ is relative prime with $n$), the coefficient of $\alpha'^0$ is the same as that of $\alpha^0$ so that $f$ is unchanged. In fact reversal is even implied by doubling modulo 5.
This construction of course generalize to all prime $n$, and in fact we have the closed form $$f(x_1,\dots,x_n)=\sum_{i_1+\dots+i_n=0\pmod n} x_{i_1}\dots x_{i_n}$$ which has $n(n-1)$ symmetries corresponding to the invertible affine maps $z\mapsto az+b$ of $\mathbb Z/n\mathbb Z$ (such maps preserve the condition $i_1+\dots+i_n=0 \pmod n$), so that the maximum cardinality of the images of $f$ is at most $(n-2)!$ (so for $n=3$, this degenerates to a completely symmetric function that is of no use to actually solving the equation because it doesn't give us any relation between the roots).