@MarceloNogueira: for what value of $C_{1/2}$ is $C_{1/2}K^{1/2}>K+1$ for all $K$?

Our comments overlapped, we had been typing at the same time. I had also noticed that among the aforementioned small groups, in the vast majority of them (all but 23), the square root counting function attains its maximum at a central element. Your argument offers an explanation of this observation.

Are there any general results regarding Frobenius-Schur indicators of element centralisers in finite groups? In particular, are there any infinite families of groups that may have symplectic representations but for which your criterion kicks in?

That's nice! A data point, in particular in connection with my related question (link in my answer): out of the 1911 groups of order less than 150 that have a symplectic representation, in 236 the square root counting function takes its maximum at the identity. Out of those, 8 satisfy your criterion, i.e. there is no $y$ and $\mu\in {\rm Irr}_{C_G(y)}$ with Frobenius-Schur indicator $-1$ and such that $y{\rm ker}\mu$ has order $2$ in $C_G(y)/{\rm ker}\mu$. All 8 are groups of order $128$.

"I'm only interested in writing every (not sufficiently large) $n\in\mathbb{N}$ in a desired form". But why? That is the question that several people, including myself, would like to know the answer to. What would it teach you about numbers if you knew that some particular a priori obviously fairly dense set contains all integers, rather than all with, say, one exception?

Even in the context of explicit class field theory, it is a very strange statement that it has no significance today. That programme did not get far, but it gave us CM theory, Heegner points, and with that some of the most spectacular successes of 20th century number theory, such as Gauss's class number 1 problem for imaginary quadratic fields and cases of the BSD.

I don't think the first sentence "Langlands' conjectures originally apply to the general linear group, but later included reductive groups in general" is accurate. Already in his famous letter to Weil, Langlands formulates his programme in the generality of reductive groups.

It immediately follows from Chris's comment and Merel's theorem that given K, there can only be finitely many such $p$, even as $E$ varies.